# Surfaces and other Peano Continua with no Generic Chains

Gianluca Basso  Gianluca Basso, Dipartimento di Matematica “Giuseppe Peano”,
Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
<EMAIL_ADDRESS>https://gianlucabasso.com ,  Alessandro Codenotti
Alessandro Codenotti, Dipartimento di Matematica, Universitá di Bologna,
Piazza di Porta S. Donato 5, 40126 Bologna, Italy
<EMAIL_ADDRESS>and  Andrea Vaccaro  Andrea Vaccaro,
Mathematisches Institut, Fachbereich Mathematik und Informatik der Universität
Münster, Einsteinstrasse 62, 48149 Münster, Germany<EMAIL_ADDRESS>https://sites.google.com/view/avaccaro

###### Abstract.

The space of chains on a compact connected space encodes all the different
ways of continuously growing out of a point until exhausting the space. A
chain is _generic_ if its orbit under the action of the underlying
homeomorphism group is comeager. In this paper we show that a large family of
topological spaces do not have a generic chain: in addition to all manifolds
of dimension at least 3, for which the result was already known, our theorem
covers all compact surfaces except for the sphere and the real projective
plane – for which the question remains open – as well as all other homogeneous
Peano continua, circle excluded. If the spaces are moreover strongly locally
homogeneous, which is the case for any closed manifold and the Menger curve,
we prove that chains cannot be classified up to homeomorphism by countable
structures, and that the underlying homomorphism groups have non-metrizable
universal minimal flows, in contrast to the case of 1-dimensional manifolds.
The proof of the main result is of combinatorial nature, and it relies on the
creation of a dictionary between open sets of chains on one side, and walks on
finite connected graphs on the other.

###### 2020 Mathematics Subject Classification:

Primary 37B45, Secondary 03E15, 05C38, 54F16

G.B. was partially supported by Istituto Nazionale di Alta Matematica
“Francesco Severi”. A.C. and A.V. were supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy EXC 2044–-390685587, Mathematics Münster:
Dynamics–Geometry–Structure, through SFB 1442. A.C. was moreover supported by
the ERC Starting Grant 101077154 “Definable Algebraic Topology”. A.V. was
moreover supported by the ERC Advanced Grant 834267–AMAREC

##  1\. Introduction

Given a metrizable compact connected space $X$, a _maximal chain of compact
connected sets of $X$_ – just _chain_ , from now on –, is a collection of
compact connected subsets of $X$ which is linearly ordered by inclusion and
maximal with respect to this property. Equivalently, it is a homeomorphic
image of the interval $[0,1]$ in the hyperspace $C(X)$ of compact connected
subsets of $X$ such that $0$ is mapped to a singleton, $1$ to the point $X\in
C(X)$ and for each $s<t$, the image of $s$ is a subset of the image of $t$
[illanes1999hyperspaces]*Lemma 14.7. The collection $\Phi(X)$ of all such
chains therefore represents all the different ways one can start, say at time
$0$, from a point of $X$, and then continuously grow out of it, until, at time
$1$, they have exhausted $X$.

A classification of such chains up to homeomorphisms of $X$ is only feasible
for very simple spaces, such as the the interval or the circle (see Appendix
A). At the very least, we can ask for which spaces there is _essentially_ only
one chain, that is, whether there is $\mathcal{C}\in\Phi(X)$ such that its
orbit under the action $\operatorname{Homeo}(X)\curvearrowright\Phi(X)$ is
comeager in $\Phi(X)$, endowed with its natural compact hyperspace topology.
In such case, we say that $\mathcal{C}$ is a _generic chain_.

Focusing on compact manifolds, the chain on the circle consisting of all
intervals centered around some point $x_{0}$ is generic in $\Phi(S^{1})$, and
a similar statement holds for the closed interval (see Proposition A.4 and
Theorem A.6). On the other hand, Gutman, Tsankov and Zucker proved in [GTZ]
that no closed manifold of dimension at least $3$ has generic chains, and nor
does the Hilbert cube. The case of surfaces was left open. A corollary of our
main theorem gives the following.

######  Theorem 1.1.

If $X$ is a compact surface other than the sphere or the real projective
plane, then $X$ has no generic chain.

We obtain Theorem 1.1 as consequence of a stronger statement which not only
recovers the aforementioned result of [GTZ], but which moreover covers a vast
class of Peano continua going well beyond the context of manifolds.

A _Peano continuum_ is a metrizable compact connected space which is locally
connected. Among notable examples which are not manifolds are the Sierpiński
carpet and the Menger curve. These are universal one-dimensional continua
obtained by iteratively carving out the square and the cube, respectively,
similarly to how the Cantor space is obtained from the unit interval.

Before stating our main theorem, we introduce some basic terminology. A subset
$K\in X$ is _locally separating_ if $U\setminus K$ is disconnected for some
connected open $U\subseteq X$. A space is _planar_ if it admits an embedding
in $\mathbb{R}^{2}$ and it is _locally non-planar_ if none of its nonempty
open sets are planar. Let us say that a Peano continuum $X$ _has a circular
covering_ if it admits a covering $O_{0},\dots,O_{\ell-1}$ in $\ell\geq 4$
_connected_ open subsets such that
$\operatorname{Cl}(O_{i})\cap\operatorname{Cl}(O_{j})\neq\emptyset$ if and
only if $\lvert i-j\rvert\leq 1\text{ mod}\,\ell$, and
$O_{i}\setminus\bigcup_{j\neq i}O_{j}$ is connected for each $i<\ell$.

Our main result is the following.

######  Theorem 1.2.

Let $X$ be a Peano continuum with no locally separating points. If $X$ either

  1. (1)

has a locally non-planar open subset,

  2. (2)

has a planar open set containing a simple closed curve which is not locally
separating, or

  3. (3)

has a circular covering,

then $X$ has no generic chain.

Let us parse the above hypotheses for $X$ a compact manifold. A compact
manifold has a locally separating point if and only if it is $1$-dimensional,
and it has a locally non-planar open subset if and only if it is at least
$3$-dimensional. A boundary of a surface is a simple closed curve which is not
locally separating and lies in a planar open set. Finally, the sphere and the
real projective plane are the only closed surfaces without a circular covering
(see Proposition 6.13).

We prove Theorem 1.2 by first establishing a correspondence between Peano
continua and open sets of chains on one side, and finite connected graphs and
walks on these graphs on the other. We then find a combinatorial necessary
condition for the existence of a generic chain on a Peano continuum, and prove
that it is does not hold under our assumptions.

The combinatorial condition is an _off-by-one_ weak amalgamation principle,
roughly stating that each walk can be refined by another walk in such a way
that any two different walks refining it can be reconciled by a small
perturbation. We extract the combinatorial idea at the core of the main proof
of [GTZ] – that chains can be made irreconcilable by winding enough times in
opposite directions – and discretize it, producing irreconcilable walks which
wind around circular subgraphs.

The discretization relies on a handful of classical continuum-theoretical
black-boxes, all of which are stated in Section 4. In the following two
sections, these black-boxes are used to derive basic combinatorial statements
about graphs and walks on graphs. The proof of Theorem 1.2 is purely
combinatorial.

This approach to the study of topological and dynamical properties of compact
metrizable spaces by combinatorial means traces its roots to the field of
_projective Fraïssé theory_ , first introduced in [Irwin2006]. In rough terms,
one constructs an infinite compact zero-dimensional graph, the _combinatorial
model_ of $X$, and finite approximations of such graph, and studies them to
derive properties about $X$, leaning on the dictionary between the
combinatorics of countable structures and the dynamics of Polish non-
archimedean groups put forth in the seminal paper [KPT]. The aforementioned
Menger curve was studied under this lens in [PS22], and it follows from the
results in [BK:Lelek], again obtained in a projective Fraïssé theoretical
framework, that the Lelek fan, a one-dimensional continuum which is not
locally connected, has a generic _downward closed_ chain.

Nonetheless, our strategy is innovative in that it is the first instance in
which non-trivial _necessary_ conditions for the existence of a comeager orbit
have been translated to combinatorial statements. In this paper, we adopt a
minimalistic approach and we restrict ourselves to finite combinatorial
objects, though our proofs could also be used to derive the analogous
statements for the combinatorial models of the spaces involved.

Theorem 1.2 has important dynamical consequences for the groups of
homomorphisms involved. If $G$ is a topological group, we call a continuous
action $G\curvearrowright Y$ on a compact space a _$G$ -flow_. A $G$-flow is
_minimal_ if every orbit is dense. Both classifying minimal flows of a group
$G$ and, viceversa, classifying groups based on the properties of their
minimal flows are central goals in _topological dynamics_. Some groups of
homeomorphisms have very simple minimal dynamics: the only minimal flow of the
group $\operatorname{Homeo}_{+}([0,1])$ of orientation preserving
homeomorphisms of the interval is the trivial action and the only non-trivial
minimal flow of $\operatorname{Homeo}_{+}(S^{1})$ is the action on the circle
itself (see [Pestov]).

A number of dividing lines regarding the complexity of the minimal dynamics of
a topological group have been studied, and are particularly well understood
when $G$ is Polish, i.e., separable and completely metrizable. A Polish group
is _extremely amenable_ if all of its minimal flows are trivial, it is _CAP_
if they are metrizable (see [CAP]), and it has the _generic point property_ if
every minimal flow has a comeager orbit (see [AKL]). Every extremely amenable
group is CAP, every Polish CAP group has the generic point property (see
[ben2017metrizable]), and no locally compact group has the generic point
property. Such properties can also be stated in terms of the _universal
minimal flow_ of $G$: CAP groups are also known as groups _with metrizable
universal minimal flow._

Until recently (see [kaleido]), there was essentially only one known example
of Polish group with the generic point property which is not CAP (see
[kwiat]). In [GTZ] it is proved that the subgroup
$\operatorname{Homeo}_{0}(X)$ of homeomorphisms isotopic to the identity is
not CAP, for any closed surface $X$, but it was left open whether such groups
have the generic point property. Since for a closed surface $X$ the flow
$\operatorname{Homeo}_{0}(X)\curvearrowright\Phi(X)$ is minimal
([gutman_minimal]), Theorem 1.2 allows us to cover all but two of these cases.

######  Theorem 1.3.

If $X$ is a closed surface which is not the sphere or the real projective
plane, then $\operatorname{Homeo}_{0}(X)$ does not have the generic point
property.

The following fascinating questions remain open.

######  Question 1.4 ([GTZ]*Question 1.3).

Is there a generic chain on the sphere $S^{2}$? Does
$\operatorname{Homeo}(S^{2})$ have the generic point property? What about the
real projective plane?

Let us end this introduction with a series of corollaries of the Theorem 1.2.
A space $X$ is _homogeneous_ if the action
$\operatorname{Homeo}(X)\curvearrowright X$ is transitive. Homogeneous Peano
continua are either the circle, or a closed surface, or are locally non-planar
(see [MR2674033]*Remark 8).

######  Corollary 1.5.

No homogeneous Peano continuum except for the circle and, possibly, the sphere
and real projective plane, has a generic chain.

An example of a homogeneous Peano continuum is the Menger curve, which is the
unique one-dimensional locally non-planar Peano continuum without locally
separating points (see [MR0096180]). Another is the Hilbert cube, for which
the non-existence of generic chain was already proved in [GTZ].

On the other hand, the Sierpiński carpet, which is the unique one-dimensional
planar Peano continuum without locally separating points (see [whyburn]), is
not homogeneous. Nonetheless, it has plenty of non locally separating simple
closed curves, since it can be constructed as the complement, in the sphere,
of countably many pairwise disjoint open disks whose union is dense in the
sphere itself. In particular, any open set is planar and contains a non
locally separating simple closed curve, so the Sierpiński carpet falls within
the scope of Theorem 1.2.

A one-dimensional Peano continuum with no locally separating points is either
the Menger curve or has a nonempty open connected subset homeomorphic to an
open subset of the Sierpiński carpet (see [MR1314981]*Lemma 1.1). Therefore

######  Corollary 1.6.

One-dimensional Peano continua without locally separating points do not have a
generic chain.

We prove in Theorem 7.7 that a sufficient condition for minimality of the
action $\operatorname{Homeo}(X)\curvearrowright\Phi(X)$, when $X$ is a Peano
continuum without locally separating points, is for $X$ to be _strongly
locally homogeneous_ , see Section 7 for the definition. For example, all
closed manifolds are strongly locally homogeneous. Since these spaces are in
particular homogeneous, they have no generic chain, by Corollary 1.5, except
possibly if they are the sphere or the real projective plane. Therefore:

######  Corollary 1.7.

If $X$ is a strongly locally homogeneous Peano continuum which is not the
circle, the sphere, or the real projective plane, then
$\operatorname{Homeo}(X)$ does not have the generic point property. In
particular, its universal minimal flow is not metrizable.

Among other examples of spaces which fit the hypotheses of Corollary 1.7 are
the Menger curve, the universal $k$-dimensional compactum $\mu^{k}$, and any
compact connected $\mu^{k}$-manifold, for $k\geq 1$ (see [MR0920964]*Theorem
3.2.2).

For the spaces in Corollary 1.7, we are also able to prove that the action
$\operatorname{Homeo}(X)\curvearrowright\Phi(X)$ is _turbulent_ , which
implies a strong non-classifiability result for the orbit equivalence relation
of the action. We say that chains on $X$ are _classifiable by countable
structures_ if there is a Borel function
$\mathcal{M}\colon\Phi(X)\to\mathrm{Mod}(\mathcal{L})$ to the space of
structures with domain $\mathbb{N}$ in some countable relational language
$\mathcal{L}$ such that

$\mathcal{C}^{\prime}\in\operatorname{Homeo}(X)\cdot\mathcal{C}\Longleftrightarrow\mathcal{M}(\mathcal{C})\cong\mathcal{M}(\mathcal{C}^{\prime}),$

that is, if we can reduce the orbit equivalence relation among chains to the
isomorphism relation between the associated structures. By Hjorth’s theorem
(see [hjorth2000classification]) we obtain:

######  Corollary 1.8.

If $X$ is a strongly locally homogeneous Peano continuum which is not the
circle, the sphere, or the real projective plane, then chains on $X$ are not
classifiable by countable structures.

This is in contrast with the case of $1$-dimensional manifolds, whose chains
are classifiable by countable structures and have a generic element, as we
prove in Appendix A. The techniques in this appendix are distinct from those
in the other sections.

We remark that the first part of 1.4 could be answered in the positive by
proving that chains on the sphere are classifiable by countable structures,
but we believe that this is not the case.

In Appendix B, we prove that $\operatorname{Homeo}(S)\curvearrowright\Phi(S)$
is minimal, for the Sierpiński carpet $S$, and thus that
$\operatorname{Homeo}(S)$ does not have the generic point property, and its
universal minimal flow is not metrizable. The proof is rather lengthy and
technical; it could foreseeably be simplified by finer results on the degree
of transitivity of the diagonal actions
$\operatorname{Homeo}(S)\curvearrowright S^{n}$, which we were not able to
find in the literature. We leave the question of classifiability by countable
structures of chains on the Sierpiński carpet open.

###  1.1. Plan of the paper

Sections 2 and 3 are devoted to preliminaries on chains and walks on graph,
respectively. In section 4 we recall Bing’s definition of brick partitions of
Peano continua and state Bing’s theorem – which serves as our main
discretization tool – and other relevant facts about them.

In section 5 we construct the dictionary between open sets of chains of Peano
continua and walks on graphs, and prove the required discretization results to
establish the combinatorial necessary condition for the existence of a generic
chain, which is Theorem 5.7. Section 6 is the heart of the paper and contains
the proof of Theorem 1.2, which we obtain as a sum of Theorems 6.4, 6.6, 6.10,
and 6.12, whose proofs are completely combinatorial. Section 7 is devoted to
the question of minimality and turbulence of
$\operatorname{Homeo}(X)\curvearrowright\Phi(X)$, and their consequences for
classifiability of chains and topological dynamics of
$\operatorname{Homeo}(X)$.

Finally, appendices A and B are devoted to the cases of $1$-dimensional
manifolds, and of the Sierpiński carpet, respectively.

### Acknowledgements

We are grateful to Riccardo Camerlo, Aleksandra Kwiatkowska, Aristotelis
Panagiotopoulos, and Todor Tsankov for helpful discussions at different stages
of the project.

##  2\. Peano Continua and Chains

In this preliminary section we briefly introduce Peano continua and their
spaces of maximal connected chains. We refer to [nadler2017continuum,
illanes1999hyperspaces] for a more thorough presentation of these topics.

A _continuum_ is a compact connected metrizable space. A _Peano continuum_ is
a locally connected continuum. An _arc_ is a homeomorphic image of the
interval $[0,1]$; a _continuum theoretical graph_ is a set composed of
finitely many arcs that intersect each other at most in their endpoints; a
_continuum theoretical tree_ is a connected continuum theoretical graph that
contains no _simple closed curve_ , meaning no homeomorphic copy of $S^{1}$.

We refer the reader to [nadler2017continuum]*Chapter VIII for the proofs of
the following basic properties of Peano continua which will be implicitly used
throughout the rest of the paper.

  1. (1)

Every Peano continuum is arcwise connected.

  2. (2)

Every open connected subset of a Peano continuum is arcwise connected.

  3. (3)

Whenever $U$ is an open connected subset of a Peano continuum and $F\subseteq
U$ is a finite set of points, there exists a continuum theoretical tree
$T\subseteq U$ with $F\subseteq T$.

Given a topological space $X$ and $Y\subseteq X$ we denote by
$\operatorname{Cl}(Y)$, $\operatorname{Int}(Y)$ and $\operatorname{Bd}(Y)$
respectively the closure, interior and boundary of $Y$.

Given a family $\mathcal{W}$ of open subsets of a space $X$ and $C\subseteq
X$, we define

$\operatorname{Star}(C,\mathcal{W})\coloneqq\\{W\in\mathcal{W}:\operatorname{Cl}(W)\cap
C\not=\emptyset\\}.$

It is generally more common to define $\operatorname{Star}(C,\mathcal{W})$ as
the set of those $W\in\mathcal{W}$ that meet $C$, without taking their
closures first. Our definition is slightly different, but it will simplify the
notation throughout the paper.

Let $\mathcal{V}$ be another family of open subsets of $X$ and, for
$V\in\mathcal{V}$, let

$\mathcal{W}(V)\coloneqq\\{W\in\mathcal{W}:W\subseteq V\\},$

and

$\operatorname{Core}(V,\mathcal{W})\coloneqq\\{W\in\mathcal{W}:\operatorname{Cl}(W)\subseteq
V\\}.$

Note that if $\mathcal{W}$ is a family of connected open sets, then
$\operatorname{Core}(V,\mathcal{W})=\mathcal{W}(V)\setminus\operatorname{Star}(\operatorname{Bd}(V),\mathcal{W})$.

An open set $U$ is _regular_ if $U=\operatorname{Int}(\operatorname{Cl}(U))$.
If $U,V\subseteq X$ are regular open sets, let $U\vee V$ be the regular open
set $\operatorname{Int}(\operatorname{Cl}(U)\cup\operatorname{Cl}(V))$ and,
similarly, let
$\bigvee\mathcal{U}\coloneqq\operatorname{Int}(\bigcup_{U\in\mathcal{U}}\operatorname{Cl}(U))$,
for any family $\mathcal{U}$ of regular open sets.

Let $\operatorname{Homeo}(X)$ denote the set of all homeomorphisms from $X$
onto itself. The _compact-open topology_ on $\operatorname{Homeo}(X)$ is the
topology generated by sets of the following form, with $K$ being a compact
subset of $X$ and $U$ an open one:

$\\{f\in\operatorname{Homeo}(X):f[K]\subseteq U\\}.$

If $X$ is a compact metrizable space, the compact-open topology on
$\operatorname{Homeo}(X)$ agrees with the topology of uniform convergence and
turns $\operatorname{Homeo}(X)$ into a Polish topological group.

###  2.1. The Space of Maximal Connected Chains

Given a regular topological space $X$, let $K(X)$ denote the set of compact
subspaces of $X$. The _Vietoris topology_ on $K(X)$ is generated by sets
obtained from nonempty regular open subsets $U_{0},\dots,U_{n-1}$ of $X$ as

$O(U_{0},\dots,U_{n-1})\coloneqq\left\\{K\in K(X):K\cap
U_{i}\not=\emptyset\text{ for every }i<n\text{ and
}K\subseteq\bigvee_{i<n}U_{i}\right\\}.$

If $X$ is compact, the Vietoris topology on $K(X)$ is also compact and the
subspace $C(X)\subseteq K(X)$ of _subcontinua_ , i.e. compact connected sets,
is closed (hence compact itself). If $X$ is locally connected, a basis for
this topology on $C(X)$ is given by sets of the form $O(U_{0},\dots,U_{n-1})$,
where $U_{i}$ are open and connected and such that $\bigvee_{i<n}U_{i}$ is
also connected.

A collection $\mathcal{C}$ of compact subsets of $X$ is called a _chain_ (a
_nest_ in [illanes1999hyperspaces]) if it is linearly ordered by inclusion,
that is, if for every $C_{1},C_{2}\in\mathcal{C}$, either $C_{1}\subseteq
C_{2}$ or $C_{2}\subseteq C_{1}$. A chain $\mathcal{C}\subseteq K(X)$ is
_maximal_ if it is not properly contained in any other chain and it is
_connected_ if it consists of connected sets.

If $X$ is compact and $\mathcal{C}\subseteq K(X)$ is a maximal chain, then
$\mathcal{C}$ is compact (as a subspace of $K(X)$), hence it can be thought of
as a point in $K(K(X))$. If moreover $X$ is a continuum, every maximal
connected chain is the image of a topological embedding $[0,1]\to C(X)$, so it
is in particular an element of $C(C(X))$ (see [illanes1999hyperspaces]*Lemma
14.7). In this case, we define the space $\Phi(X)$ of _maximal connected
chains on $X$_ as

$\Phi(X)\coloneqq\\{\mathcal{C}\in C(C(X)):\mathcal{C}\text{ is a maximal
connected chain}\\},$

which is a compact subset of $C(C(X))$ (see [uspenskij2000universal]*§2). The
map $\Phi(X)\to X$ that that to each chain $\mathcal{C}$ associates the unique
element of the singleton $\bigcap\mathcal{C}$ – the so called _root_ of
$\mathcal{C}$ – is continuous.

_From now on, all chains we consider are maximal and connected._

A $\pi$_-basis_ $\mathcal{B}$ of a topological space $X$ is a collection of
open subsets of $X$ such that for every open $U\subseteq X$ there exists
$B\in\mathcal{B}$ with $B\subseteq U$. Given a $\pi$-basis $\mathcal{B}$ of
$X$, one can construct a $\pi$-basis of $\Phi(X)$ by considering sets of the
form described below as $\mathcal{B}_{0},\dots,\mathcal{B}_{n-1}$ vary among
all finite subsets of $\mathcal{B}$:

$\Phi(X)\cap\bigcap_{i<n}O\left(O(\mathcal{B}_{0}),\dots,O(\mathcal{B}_{i})\right)=\\{\mathcal{C}\in\Phi(X):\mathcal{C}\cap
O(\mathcal{B}_{i})\neq\emptyset\text{ for every }i<n\\}.$

The action $\operatorname{Homeo}(X)\curvearrowright X$ naturally induces a
continuous action $\operatorname{Homeo}(X)\curvearrowright\Phi(X)$ defined as
$h\cdot\mathcal{C}\coloneqq\\{h[C]:C\in\mathcal{C}\\}$, for
$h\in\operatorname{Homeo}(X)$ and $\mathcal{C}\in\Phi(X)$. We say that a chain
$\mathcal{C}\in\Phi(X)$ is _generic_ if
$\operatorname{Homeo}(X)\cdot\mathcal{C}$ is comeager in $\Phi(X)$.

##  3\. Walks on Graphs

Throughout this paper, the word graph refers to finite connected reflexive
graphs.

In this section we present definitions and terminology on graphs and walks, as
well as some preliminary results on the notion of winding number, which is
introduced in Definition 3.2.

Let $(G,E)$ be a graph. We denote by $d_{G}$, or just $d$ if there is no risk
of confusion, the _path metric_ on $G$. If $x\in G$ and $A\subseteq G$, we
write $x\mathbin{E}A$ if there is $y\in A$ with $x\mathbin{E}y$ and we denote
by $[A]_{n}\coloneqq\\{x\in G:d_{G}(x,A)\leq n\\}$ the closed ball of
$d_{G}$-radius $n$ around $A$. If $A=\\{x\\}$, we write $[x]_{n}$.

If $f,g\colon(G,E)\to(H,E^{\prime})$ are functions between the same two
graphs, we denote by $d_{\sup}(f,g)\coloneqq\max\\{d_{H}(f(x),g(x)):x\in G\\}$
their distance according to the supremum metric.

A function $f\colon(G,E)\to(H,E^{\prime})$ is a _monotone epimorphism_ (also
called _connected epimorphism_ in [PS22]) if it is a homomorphism of graphs
such that the preimage of each nonempty connected subset of $H$ is nonempty
and connected in $G$. Equivalently, if it is a homomorphism of graphs which is
surjective on edges and such that the preimage of each vertex of $H$ is
nonempty and connected (see [PS22]*Lemma 1.1). In particular, monotone
epimorphisms are onto.

A _walk_ on $(G,E)$ is a sequence of vertices $w=\langle
w(0),\dots,w(\ell-1)\rangle$ such that $w(i)\mathbin{E}w(i+1)$ for every
$i<\ell-1$. We say that $\ell$ is the _length_ of $w$ and denote it by
$\operatorname{lh}(w)$. We admit the empty walk, which has length $0$.
Adopting a common programming language convention, let us write $w(-1)$ in
lieu of $w(\operatorname{lh}(w)-1)$ for the last element of $w$. With a slight
abuse of notation, we shall write $x\in w$ whenever $x\in G$ is such that
$w(i)=x$ for some $i<\operatorname{lh}(w)$.

If $f\colon(G,E)\to(H,E^{\prime})$ is a graph homomorphism and $w$ is a walk
on $(G,E)$, let $f\cdot w$ be the induced walk $\langle
f(w(0)),\dots,f(w(-1))\rangle$ on $(H,E^{\prime})$.

A walk $w$ is

  1. (1)

_reduced_ if $w(i)\neq w(i+1)$ for every $i<\operatorname{lh}(w)-1$;

  2. (2)

_uncrossed_ if whenever $i,j<\operatorname{lh}(w)-1$ are such that
$w(i)=w(j)$, then $w(i+1)=w(j+1)$;

  3. (3)

a _path_ if $w(i)=w(j)$ if and only if $i=j$;

  4. (4)

a _spaced path_ if $w(i)\mathbin{E}w(j)$ if and only if $|i-j|\leq 1$ for
every $i,j<\operatorname{lh}(w)$.

If $A,B\subseteq C\subseteq G$ are nonempty subsets, a _path from $A$ to $B$
in $C$_ is a path $w$ on $G$ such that $w(0)\in A,w(-1)\in B$ and $w(i)\in
C\setminus(A\cup B)$ for all $0<i<\operatorname{lh}(w)-1$. Such a path always
exists if $C$ is connected. If $w$ is a path, we refer to $w(0),w(-1)$ as its
_endpoints_.

A walk is _circular_ if $w(-1)=w(0)$. We modify the definitions of path/spaced
for circular walks accordingly. We also modify the definition of length, so
that $\operatorname{lh}(w)=\ell$ if $w=\langle
w(0),\dots,w(\ell-1),w(0)\rangle$. This is off-by-one with respect to its
length as a walk.

Given two walks $w,v$ on $(G,E)$, we say that $v$ is _confined_ in $w$ if

$\\{v(j):j<\operatorname{lh}(v)\\}\subseteq\\{w(i):i<\operatorname{lh}(w)\\}$

and that it is _confined_ in some $A\subseteq G$ if
$\\{v(j):j<\operatorname{lh}(v)\\}\subseteq A$. We say that $w$ _refines_ $v$,
in symbols $w\sqsubseteq v$, if:

for every $s\leq\operatorname{lh}(v)$, there is $t\leq\operatorname{lh}(w)$
such that $\\{w(i):i<t\\}=\\{v(j):j<s\\}$.

In words: $w$ starts on the same vertex as $v$, it then visits each _new_
vertex that $v$ visits, in the same order, possibly backtracking from time to
time, until it covers all that has been visited by $v$, after which it is
allowed to roam free.

We say that $w$ _monotonically_ refines $v$ if there is a strictly increasing
function $f\colon\operatorname{lh}(v)\to\operatorname{lh}(w)$ with $f(0)=0$
and such that

$w(i)=v(j)$ for all $f(j)\leq i<f(j+1)$ when $j<\operatorname{lh}(v)-1$,

and $w(i)=v(-1)$ for $i\geq f(\operatorname{lh}(v)-1)$. If $w$ monotonically
refines $v$, then it refines $v$, and $w,v$ are confined to one another.

Given two walks $w,v$ on $(G,E)$ such that $w(-1)\mathbin{E}v(0)$, let
$w{}^{\frown}v$ be the walk obtained by the _concatenation_ of $w$ and $v$.

Finally, we will say that a walk is a _lasso_ if it can be written as
$v{}^{\frown}c$, where $v$ is a walk and $c$ is a circular walk; it is a
(spaced) lasso path if moreover $v$ is a (spaced) path and $c$ is a (spaced)
circular path disjoint from $v$. When we say that a lasso $v{}^{\frown}c$
monotonically refines another $v^{\prime}{}^{\frown}c^{\prime}$, we
additionally require that $v$ is confined to $v^{\prime}$ and $c$ to
$c^{\prime}$.

Monotone maps allow to lift walks, as we see in the following lemma.

######  Lemma 3.1.

Let $f\colon(G,E)\to(H,E^{\prime})$ be a monotone epimorphism between graphs,
and let $w$ be a reduced walk on $(H,E^{\prime})$. Then there is a reduced
walk $v$ on $(G,E)$ such that $f\cdot v$ monotonically refines $w$. Moreover,
$v$ can be chosen to be a path and/or circular/lasso, whenever $w$ is.

###### Proof.

Let $P_{0}$ be a path of length $1$ in $f^{-1}(w(0))$, and choose it moreover
to be $E$-adjacent to $f^{-1}(w(-1))$ if $w$ is circular. Suppose we have
defined $P_{i}$, for $i<\operatorname{lh}(w)-1$, such that $P_{i}(-1)\in
f^{-1}(w(i))$, and find a path $P_{i+1}$ from $P_{i}(-1)$ to $f^{-1}(w(i+1))$
in $f^{-1}(w(i))\cup f^{-1}(w(i+1))$, which is possible since the latter set
is connected. Notice that all of $P_{i+1}$ except its last element is in
$f^{-1}(w(i))$.

Adapt the last step accordingly if $w$ is a lasso or circular: in this latter
case set the only element of $P_{0}$ to be the final endpoint of $P_{i+1}$.
The reduced concatenation of the $P_{i}$’s is the required walk $v$. If $w$ is
a path, then the $P_{i}$’s meet at most in their endpoints, so $v$ is again a
path. ∎

###  3.1. The Winding Number of a Walk

######  Definition 3.2.

Let $(G,E)$ be a graph and $C$ be a circular path on $G$ of length $\ell$. For
each ordered pair of vertices $(x,y)$ in $G$, we define its _weight_ relative
to $C$ to be:

$\operatorname{weight}_{C}(x,y)\coloneqq\begin{cases}1&\text{if there is
}i<\ell\text{ such that }x=C(i),\,y=C(i+1\text{ mod}\,\ell)\\\ -1&\text{if
there is }i<\ell\text{ such that }y=C(i),\,x=C(i+1\text{ mod}\,\ell)\\\
0&\text{otherwise.}\end{cases}$

Notice that $\operatorname{weight}_{C}$ is well defined since $C$ is a path.
For any walk $w$ on $G$, we define its _winding number around $C$_ as

$\operatorname{wd}_{C}(w)\coloneqq\sum_{i<\operatorname{lh}(w)-1}\operatorname{weight}_{C}(w(i),w(i+1)).$

We devote the end of the section to a few easy lemmas on walks and their
winding numbers which will be applied later in the paper. The impatient reader
is advised to skip them and return to consult them during the proof of Theorem
6.4. The keen reader, on the other hand, is encouraged to compare Lemma 3.4
with [GTZ]*Lemma 2.12 and Lemma 3.5 with [GTZ]*Lemma 2.10. For ease of
reference, the graphs are called $\mathcal{S}$ and $\mathcal{W}$ rather than
$(G,E)$ or $(H,E^{\prime})$, as elsewhere in this section.

######  Lemma 3.3.

Let $\mathcal{S}$ be a graph and $C$ be a circular path on $\mathcal{S}$. Let
$v$ and $w$ be walks on $\mathcal{S}$ such that $w$ monotonically refines $v$.
Then $\operatorname{wd}_{C}(w)=\operatorname{wd}_{C}(v)$.

###### Proof.

By assumption there is a strictly increasing function
$f\colon\operatorname{lh}(v)\to\operatorname{lh}(w)$ with

$f(0)=0\text{ and such that }w(i)=v(j)\text{ for all }f(j)\leq i<f(j+1).$

In particular $\operatorname{weight}_{C}(w(i),w(i+1))=0$ whenever there is
$j<\operatorname{lh}(v)$ with $f(j)\leq i<i+1<f(j+1)$, since in this case
$w(i)=w(i+1)$. Therefore:

$\displaystyle\operatorname{wd}_{C}(w)$
$\displaystyle=\sum_{j<\operatorname{lh}(w)-1}\operatorname{weight}_{C}(w(f(j)),w(f(j+1)))$
$\displaystyle=\sum_{j<\operatorname{lh}(v)-1}\operatorname{weight}_{C}(v(j),v(j+1))$
$\displaystyle=\operatorname{wd}_{C}(v).$

∎

######  Lemma 3.4.

Let $\mathcal{S}$ be a graph and $C$ be a circular path on $\mathcal{S}$. Let
$\mathcal{W}$ be a graph and $\alpha\colon\mathcal{W}\to\mathcal{S}$ be a
graph homomorphism. Let $w$ be a spaced path on $\mathcal{W}$ and $z$ be a
walk on $\mathcal{W}$ which refines and is confined to an initial segment of
$w$. Then

$\operatorname{wd}_{C}(\alpha\cdot z)=\operatorname{wd}_{C}(\alpha\cdot\langle
w(0),\dots,w(j)\rangle),$

where $j<\operatorname{lh}(w)$ is the unique index such that $w(j)=z(-1)$.

###### Proof.

We proceed by induction on the length $\ell$ of $z$. If $\ell=1$, then

$\operatorname{wd}_{C}(\alpha\cdot
z)=0=\operatorname{wd}_{C}(\alpha\cdot\langle w(0)\rangle).$

For $\ell>1$, suppose that the statement is true for $\langle
z(0),\dots,z(\ell-2)\rangle$. Suppose $z(\ell-2)=w(i)$, for some
$i<\operatorname{lh}(w)$. Since $z$ refines and is confined to an initial
segment of $w$, which is a spaced path, it must be that $z(\ell-1)$ is equal
to either $w(i-1),w(i)$, or $w(i+1)$. In each case, a straightforward
computation gives the desired result. ∎

######  Lemma 3.5.

Let $\mathcal{S}$ be a graph and $C$ be a spaced circular path on
$\mathcal{S}$ of length $\ell\geq 4$. Let $z,z^{\prime}$ be walks of the same
length $k$ on $\mathcal{S}$, which are confined to $C$ and such that, for each
$i<k$, $d_{\mathcal{S}}(z(i),z^{\prime}(i))\leq 1$, and also, for $i<k-1$,
$d_{\mathcal{S}}(z^{\prime}(i),z(i+1))\leq 1$. Then
$\lvert\operatorname{wd}_{C}(z)-\operatorname{wd}_{C}(z^{\prime})\rvert\leq
2$.

###### Proof.

We prove that under the hypotheses:

$\operatorname{wd}_{C}(z^{\prime})=\operatorname{weight}_{C}(z^{\prime}(0),z(0))+\operatorname{wd}_{C}(z)+\operatorname{weight}_{C}(z(-1),z^{\prime}(-1)).$

The conclusion then follows immediately, since the values
$\operatorname{weight}_{C}(z^{\prime}(0),z(0))$ and
$\operatorname{weight}_{C}(z(-1),z^{\prime}(-1))$ have absolute values at most
$1$.

We show by induction that for each $i<k$:

(3.1) $\operatorname{wd}_{C}(\langle
z^{\prime}(0),\dots,z^{\prime}(i)\rangle)=\operatorname{weight}_{C}(z^{\prime}(0),z(0))+\operatorname{wd}_{C}(\langle
z(0),\dots,z(i)\rangle)+\operatorname{weight}_{C}(z(i),z^{\prime}(i)).$

If $i=0$, then
$\operatorname{weight}_{C}(z^{\prime}(0),z(0))=-\operatorname{weight}_{C}(z(0),z^{\prime}(0))$,
so (3.1) holds.

Suppose (3.1) is true for $i<k-1$ and let us prove it for $i+1$. By definition
of $\operatorname{wd}_{C}$ it suffices to verify that

$\operatorname{weight}_{C}(z^{\prime}(i),z^{\prime}(i+1))=\operatorname{weight}_{C}(z(i),z(i+1))+\\\
+\operatorname{weight}_{C}(z(i+1),z^{\prime}(i+1))-\operatorname{weight}_{C}(z(i),z^{\prime}(i)).$

But
$-\operatorname{weight}_{C}(z(i),z^{\prime}(i))=\operatorname{weight}_{C}(z^{\prime}(i),z(i))$
and, using that $z^{\prime}(i),z(i+1)$ lie in $C$ at distance at most $1$ we
get

$\operatorname{weight}_{C}(z^{\prime}(i),z(i))+\operatorname{weight}_{C}(z(i),z(i+1))=\operatorname{weight}_{C}(z^{\prime}(i),z(i+1)).$

Also $z^{\prime}(i),z^{\prime}(i+1)$ are at distance at most $1$, so

$\operatorname{weight}_{C}(z^{\prime}(i),z(i+1))+\operatorname{weight}_{C}(z(i+1),z^{\prime}(i+1))=\operatorname{weight}_{C}(z^{\prime}(i),z^{\prime}(i+1)),$

and we are done. ∎

##  4\. Brick Partitions

In this section we summarize some well-known definitions and facts about brick
partitions of Peano continua which will be of use in the next sections.

A _quasi-partition_ of a space $X$ is a finite family $\mathcal{W}$ of
pairwise disjoint regular open subsets whose union is dense in $X$. The _mesh_
of a quasi-partition is the maximum of the diameters of its elements. Each
quasi-partition $\mathcal{W}$ gives rise to a finite graph $(\mathcal{W},E)$ –
its _nerve graph_ –, namely the graph whose vertex set is $\mathcal{W}$ and
such that there is an edge $W\mathbin{E}W^{\prime}$ if and only if
$\operatorname{Cl}(W)\cap\operatorname{Cl}(W^{\prime})\not=\emptyset$. To
avoid confusion, we say that $A\subseteq\mathcal{W}$ is _$\mathbin{E}$
-connected_ when $A$ is connected as a subgraph of $(\mathcal{W},E)$. With a
slight abuse of notation, we shall denote the edge relation on nerve graphs of
different quasi-partitions always with the letter $E$ (it will always be clear
from the context to which quasi-partition we are referring to).

A metric space $X$ is _uniformly locally connected_ if for every
$\varepsilon>0$ there exists $\delta>0$ such that for all $x,y\in X$,
$d(x,y)<\delta$ implies that there is a connected set $C\subseteq X$ with
$x,y\in C$ and $\operatorname{diam}(C)<\varepsilon$. When $X$ is a Peano
continuum, $C$ can be assumed to be an arc.

A quasi-partition $\mathcal{W}=\\{W_{0},\ldots,W_{n-1}\\}$ of a Peano
continuum $X$ is called a _brick partition_ if $W_{i}$ is connected and
uniformly locally connected for every $i<n$, and $W_{i}\vee W_{j}$ is
uniformly locally connected for all $i,j<n$, with respect to any compatible
metric on $X$. In particular, if $W_{i}\mathbin{E}W_{j}$ then $W_{i}\vee
W_{j}$ is connected.

Given a quasi-partition $\mathcal{U}$ of a Peano continuum $X$ and a partition
$\mathcal{P}=\\{\mathcal{U}_{0},\dots,\mathcal{U}_{n-1}\\}$ of $\mathcal{U}$
into $\mathbin{E}$-connected subsets, the _amalgam_ of $\mathcal{U}$ with
respect to $\mathcal{P}$ is the quasi-partition
$\\{\bigvee\mathcal{U}_{0},\dots,\bigvee\mathcal{U}_{n-1}\\}$ of $X$. The
amalgam of a brick partition is again a brick partition.

A family of open sets $\mathcal{W}$ is said to _refine_ another one
$\mathcal{V}$, in symbols $\mathcal{W}\preceq\mathcal{V}$, if every element of
$\mathcal{W}$ is contained in (exactly) an element of $\mathcal{V}$. In the
case of brick partitions, a refinement $\mathcal{W}\preceq\mathcal{V}$ gives
rise to a monotone epimorphism from the nerve graph of $\mathcal{W}$ to that
of $\mathcal{V}$, given by mapping $W\mapsto V$ whenever $W\subseteq V$. We
denote such map by
$\rho^{\mathcal{W}}_{\mathcal{V}}\colon\mathcal{W}\to\mathcal{V}$. Given two
brick partitions $\mathcal{W},\mathcal{V}$ and two walks $w,v$ on
$\mathcal{W}$ and $\mathcal{V}$ respectively, we write
$(\mathcal{W},w)\preceq(\mathcal{V},v)$ whenever
$\mathcal{W}\preceq\mathcal{V}$ and $\rho^{\mathcal{W}}_{\mathcal{V}}\cdot
w\sqsubseteq v$.

We will make use of the following fundamental result of Bing from [bing,
bing2] implicitly whenever we need to produce arbitrarily fine brick
partitions (i.e. with arbitrarily small mesh) of a Peano continuum.

######  Theorem 4.1 ([bing, bing2]).

Any brick partition of a Peano continuum can be refined by brick partitions of
arbitrarily small mesh.

Recall that a subset $K\in X$ is _locally separating_ if there is an open
connected set $U\subseteq X$ such that $U\setminus K$ is disconnected, that a
space is _planar_ if it admits an embedding in $\mathbb{R}^{2}$ and it is
_locally non-planar_ if none of its nonempty open sets are planar. We will use
the following facts, which easily follow from the definitions.

######  Fact 4.2.

Let $X$ be a Peano continuum and let $\mathcal{W}$ be a brick partition of $X$
and $W\mathbin{E}W^{\prime}$ be in $\mathcal{W}$.

  1. (1)

There is a point
$x\in\operatorname{Cl}(W)\cap\operatorname{Cl}(W^{\prime})\cap(W\vee
W^{\prime})$;

  2. (2)

if moreover $X$ has no locally separating points, then there are two distinct
such points;

  3. (3)

if $K\subseteq X$ is a non-locally separating closed set, then $x$ can be
chosen disjoint from $K$.

We also need these facts on the existence of certain arcs.

######  Fact 4.3.

Let $X$ be a Peano continuum, $\mathcal{W}$ be a brick partition of $X$ and
let $W\in\mathcal{W}$.

  1. (1)

For any $x,y\in\operatorname{Cl}(W)$, there is an arc $\gamma\subseteq
W\cup\\{x,y\\}$ connecting $x,y$ ([nadler2017continuum]*8.32(a)).

  2. (2)

If $X$ has no locally separating points, for any
$A,B\subseteq\operatorname{Cl}(W)$, disjoint and containing exactly two points
each, there are two disjoint arcs $\gamma_{0},\gamma_{1}\subseteq W\cup A\cup
B$, connecting $A$ to $B$ ([MR1562166]*Lemma 1).

  3. (3)

If $X$ has no locally separating points and $W$ is locally non-planar, then
for any $x_{0},x_{1},y_{0},y_{1}\in\operatorname{Cl}(W)$ all distinct, there
are two disjoint arcs $\gamma_{0},\gamma_{1}\subseteq
W\cup\\{x_{0},x_{1},y_{0},y_{1}\\}$ connecting $x_{0}$ to $y_{0}$ and $x_{1}$
to $y_{1}$, respectively ([MOT]*Theorem 3.13).

  4. (4)

If $K\subseteq X$ is a non-locally separating closed set, and
$x,y\in\operatorname{Cl}(W)$, there is an arc $\lambda\subseteq
W\cup\\{x,y\\}$ disjoint from $K$ connecting $x,y$.

##  5\. Walks on Brick Partitions and their Refinements

We now apply the framework developed in Section 3 to walks on nerve graphs of
brick partitions of Peano continua. The highlight of this section is Theorem
5.7, where we show that the existence of a generic chain on $X$ implies an
_off-by-one_ weak amalgamation principle for walks on (nerve graphs of) brick
partitions of $X$. In Section 6 we shall derive our non-existence result in
Theorem 1.2 from the failure of such principle. The ingredients of in the
proof of Theorem 5.7 are a general criterion for the existence of a comeager
orbit due to Rosendal, and two discretization results: Proposition 5.2 for
open sets of chains on $X$, and Theorem 5.6 for homeomorphism of $X$.

######  Definition 5.1.

Let $X$ be a Peano continuum and $\mathcal{W}$ a brick partition on it. Given
a walk $w=\langle w(0),\dots,w(n-1)\rangle$ on $\mathcal{W}$, we define the
following open subset of $\Phi(X)$:

$\mathbf{O}_{\mathcal{W},w}\coloneqq\left\\{c\in\Phi(X):c\cap
O\left(w(0),\dots,w(i-1)\right)\not=\emptyset\text{ for every
}i<\operatorname{lh}(w)\right\\}.$

It is immediate from the definitions that if $w^{\prime},w$ are walks on a
brick partition $\mathcal{W}$, then

$\mathbf{O}_{\mathcal{W},w^{\prime}}\subseteq\mathbf{O}_{\mathcal{W},w}\quad\text{
if and only if }\quad w^{\prime}\sqsubseteq w,$

and, more in general, that whenever $\mathcal{W}\preceq\mathcal{V}$ and $w,v$
are walk on $\mathcal{W},\mathcal{V}$, respectively

(5.1)
$\mathbf{O}_{\mathcal{W},w}\subseteq\mathbf{O}_{\mathcal{V},v}\quad\text{ if
and only if }\quad\rho^{\mathcal{W}}_{\mathcal{V}}\cdot w\sqsubseteq v.$

We also take note that for each walk $w$ on $\mathcal{W}$ there is a reduced
walk $\bar{w}$ such that
$\mathbf{O}_{\mathcal{W},\bar{w}}=\mathbf{O}_{\mathcal{W},w}$.

By varying $\mathcal{U}$ and taking $u$ to be the empty walk in the next
proposition, we see in particular that open sets determined by reduced walks
form a $\pi$-basis in $\Phi(X)$.

######  Proposition 5.2.

Let $X$ be a Peano continuum, and fix a brick partition $\mathcal{U}$ and a
walk $u$ on $\mathcal{U}$. Let
$\mathbf{V}\subseteq\mathbf{O}_{\mathcal{U},u}\subseteq\Phi(X)$ be an nonempty
open set. There exist a brick partition and a reduced walk
$(\mathcal{V},v)\preceq(\mathcal{U},u)$ such that
$\mathbf{O}_{\mathcal{V},v}\subseteq\mathbf{V}$.

###### Proof.

The collection $\\{W\subseteq X:W\in\mathcal{W}\text{ for some brick partition
}\mathcal{W}\preceq\mathcal{U}\\}$ is $\pi$-basis of $X$, so, up to
restricting $\mathbf{V}$, we can suppose that there is a brick partition
$\mathcal{V}\preceq\mathcal{U}$ and finitely many subsets
$\mathcal{V}_{i}\subseteq\mathcal{V}$ so that $\mathbf{V}$ is of the form

$\mathbf{V}=\\{c\in\Phi(X):c\cap O(\mathcal{V}_{i})\neq\emptyset\text{ for
every }i<n\\}.$

Since $\mathbf{V}$ is nonempty, each $\mathcal{V}_{i}$ is nonempty and
$\mathbin{E}$-connected. Moreover, for any $i,j<n$, it must hold that either
$\mathcal{V}_{i}\subseteq\mathcal{V}_{j}$ or
$\mathcal{V}_{j}\subseteq\mathcal{V}_{i}$, as otherwise $\mathbf{V}$ would
contain no chain. Therefore, up to reordering, we might as well suppose that
$\mathcal{V}_{i}\subseteq\mathcal{V}_{j}$ whenever $i<j<n$.

Let $v$ be any reduced walk which starts in $\mathcal{V}_{0}$, and
successively covers each $\mathcal{V}_{i}$, taking care of not exiting it
before having visited each of its elements. Then
$\mathbf{O}_{\mathcal{V},v}\subseteq\mathbf{V}\subseteq\mathbf{O}_{\mathcal{U},u}$,
and thus $\rho^{\mathcal{V}}_{\mathcal{U}}\cdot v\sqsubseteq u$, by (5.1). ∎

We state a preliminary lemma which will be needed later on.

######  Lemma 5.3 (Spacing out Paths).

Let $X$ be a Peano continuum, and fix a brick partition $\mathcal{W}$ and a
path $w$ on $\mathcal{W}$. There are a brick partition
$\mathcal{Z}\preceq\mathcal{W}$ and a spaced path $z$ on $\mathcal{Z}$ such
that $\rho^{\mathcal{Z}}_{\mathcal{W}}\cdot z$ monotonically refines $w$ and
$\rho^{\mathcal{Z}}_{\mathcal{W}}([z]_{1})\subseteq w$, meaning that for each
$Z\in z$ and $Z^{\prime}\in\mathcal{Z}$ with $Z\mathbin{E}Z^{\prime}$, we have
$\rho^{\mathcal{Z}}_{\mathcal{W}}(Z^{\prime})\in w$. If $w$ is moreover
circular or a lasso, $z$ can also be chosen to be circular or a lasso,
respectively.

###### Proof.

Find $x_{0}\in w(0)$ and $x_{\operatorname{lh}(w)}\in w(-1)$. For each
$0<i<\operatorname{lh}(w)$, let
$x_{i}\in\operatorname{Cl}(w(i-1))\cap\operatorname{Cl}(w(i))\cap(w(i-1)\vee
w(i))$, which exists by 4.2.(1). For each $i<\operatorname{lh}(w)$, let
$\gamma_{i}$ be an arc in $w(i)\cup\\{x_{i},x_{i+1}\\}$ connecting
$x_{i},x_{i+1}$, which exist by 4.3.(1). Since $w$ is a path, all the
$x_{i}$’s and all the $\gamma_{i}$’s are distinct, and the $\gamma_{i}$’s
intersect at most in their endpoints.

Let $\mathcal{Z}^{\prime}\preceq\mathcal{W}$ be a brick partition fine enough
that
$[\operatorname{Star}(\gamma_{i},\mathcal{Z}^{\prime})]_{1}\subseteq\mathcal{Z}^{\prime}(w(i-1)\vee
w(i)\vee w(i+1))$ for each $i<\operatorname{lh}(w)$, and let
$P_{i}\coloneqq\operatorname{Star}(\gamma_{i},\mathcal{Z}^{\prime}(w(i)))$.
Let $\mathcal{Z}$ be an amalgam of $\mathcal{Z}^{\prime}$ which refines
$\mathcal{W}$ and contains each $\bigvee P_{i}$ for $i<\operatorname{lh}(w)$.
Then $z\coloneqq\langle\bigvee P_{0},\dots,\bigvee
P_{\operatorname{lh}(w)-1}\rangle$ is a spaced path satisfying the conclusions
of the proposition.

If $w$ is circular, we can take $x_{0}=x_{\operatorname{lh}(w)}$ in
$\operatorname{Cl}(w(-1))\cap\operatorname{Cl}(w(0))\cap(w(-1)\vee w(0))$ and
add ‘$\\!\\!\text{ mod}\,\operatorname{lh}(w)$’ where needed in the proof
above. Similarly, if $w$ is a lasso. ∎

###  5.1. A Criterion for the Absence of Generic Chains

We recall the following criterion due to Rosendal for verifying the existence
of comeager orbits. Recall that a continuous action $H\curvearrowright Y$ is
_topologically transitive_ if $H\cdot\mathbf{U}\cap\mathbf{V}\neq\emptyset$
for any nonempty open sets $\mathbf{U},\mathbf{V}\subseteq Y$.

######  Theorem 5.4 ([ben2017metrizable]*Proposition 3.2).

Let $H$ be a Polish group, let $Y$ be a Polish space and let
$H\curvearrowright Y$ be a continuous action. The following are equivalent:

  1. (1)

There is a comeager orbit in $H\curvearrowright Y$.

  2. (2)

$H\curvearrowright Y$ is topologically transitive and for any open
neighborhood $H_{0}$ of the identity in $H$ and any nonempty open set
$\mathbf{U}\subseteq Y$ there is a nonempty open
$\mathbf{V}\subseteq\mathbf{U}$ such that for any nonempty open
$\mathbf{W}_{0},\mathbf{W}_{1}\subseteq\mathbf{V}$,
$H_{0}\cdot\mathbf{W}_{0}\cap\mathbf{W}_{1}\neq\emptyset$.

The goal of this section is to obtain a combinatorial formulation of (1)
$\Rightarrow$ (2) from Theorem 5.4 when $Y=\Phi(X)$ is the space of maximal
connected chains over a Peano continuum $X$ and $H=\operatorname{Homeo}(X)$ is
its homeomorphism group. This is done in Theorem 5.7, after establishing our
discretization theorem for homeomorphism of a Peano continuum, Theorem 5.6.
The following is a generalization of [MOT]*Corollary 2.11 (where
$\mathcal{U}=\mathcal{V}$).

######  Proposition 5.5.

Let $X$ be a Peano continuum and let $\mathcal{U},\mathcal{V}$ be brick
partitions of $X$. There is an arbitrarily fine brick partition
$\mathcal{W}\preceq\mathcal{U}$ such that:

  1. (1)

$\operatorname{Core}(V,\mathcal{W})$ is nonempty and $\mathbin{E}$-connected
for every $V\in\mathcal{V}$;

  2. (2)

for every $W\in\mathcal{W}$ there are $V\in\mathcal{V}$ and
$W^{\prime}\in\operatorname{Core}(V,\mathcal{W})$ such that
$W\mathbin{E}W^{\prime}$.

###### Proof.

Up to taking a refinement we can assume that $\mathcal{U}$ is as fine as
desired. For each $V\in\mathcal{V}$, let $T_{V}$ be a continuum theoretical
tree in $V$ which meets every $U\in\operatorname{Star}(V,\mathcal{U})$. Let
$\mathcal{W}_{0}\preceq\mathcal{U}$ be a brick partition fine enough so that
$\operatorname{Star}(T_{V},\mathcal{W}_{0})\cap\operatorname{Star}(\operatorname{Bd}(V),\mathcal{W}_{0})=\emptyset$
for all $V\in\mathcal{V}$.

Given $U\in\mathcal{U}$ and $V\in\mathcal{V}$, let $\mathcal{C}^{U,V}$ be the
set of elements $W\in\mathcal{W}_{0}(U\cap V)$ for which there is a path from
$W$ to $\operatorname{Star}(T_{V}\cap U,\mathcal{W}_{0})$ in
$\mathcal{W}_{0}(U\cap V)$ which avoids
$\operatorname{Star}(\operatorname{Bd}(V),\mathcal{W}_{0})$. Note that
$\mathcal{C}^{U,V}\subseteq\operatorname{Core}(V,\mathcal{W}_{0})$, and that
$\mathcal{C}^{U,V}$ is nonempty if and only if $U\cap V\neq\emptyset$, by the
fineness assumption on $\mathcal{W}_{0}$. Let
$\mathcal{C}^{U,V}_{0},\dots,\mathcal{C}^{U,V}_{m_{U,V}-1}$ be the
$\mathbin{E}$-connected components of $\mathcal{C}^{U,V}$.

For each $U\in\mathcal{U}$, let
$\mathcal{B}^{U}\coloneqq\mathcal{W}_{0}(U)\setminus\bigcup_{V\in\mathcal{V}}\mathcal{C}^{U,V}$,
and call its $\mathbin{E}$-connected components
$\mathcal{B}^{U}_{0},\dots,\mathcal{B}^{U}_{n_{U}-1}$. We claim that if
$\mathcal{B}^{U}$ is nonempty, then for each $V\in\mathcal{V}$ and $i<n_{U}$,

(5.2) $\mathcal{B}^{U}_{i}\nsubseteq\operatorname{Core}(V,\mathcal{W}_{0}).$

Indeed, suppose that there is
$W\in\mathcal{B}^{U}_{i}\cap\operatorname{Core}(V,\mathcal{W}_{0})$. Since
$\mathcal{W}_{0}(U)$ is connected and $U\cap V\neq\emptyset$, there is a path
$P$ from $W\in\mathcal{B}^{U}_{i}$ to
$\operatorname{Star}(T_{V},\mathcal{W}_{0})$ in $\mathcal{W}_{0}(U)$. By
assumption $W\not\in\mathcal{C}^{U,V}$, so there must be a minimal
$k<\operatorname{lh}(P)$ such that
$P(k)\in\operatorname{Star}(\operatorname{Bd}(V),\mathcal{W}_{0})$. For each
$j<k$, $P(j)\in\mathcal{W}_{0}(U\cap V)$, so $P(j)\not\in\mathcal{C}^{U,V}$,
as otherwise there would be a path from $W$ to
$\operatorname{Star}(T_{V},\mathcal{W}_{0})$ in $\mathcal{W}_{0}(U\cap V)$
which avoids $\operatorname{Star}(\operatorname{Bd}(V),\mathcal{W}_{0})$. But
$P(j)\not\in\mathcal{C}^{U,V^{\prime}}$, for $V^{\prime}\neq V$, because it is
contained in $V$, so $P(j)\in\mathcal{B}^{U}$. Moreover,
$P(k)\in\mathcal{B}^{U}$, since it belongs to
$\operatorname{Star}(\operatorname{Bd}(V),\mathcal{W}_{0})$. Since
$\mathcal{B}^{U}_{i}$ is a connected component, $P(k)\in\mathcal{B}^{U}_{i}$,
so $\mathcal{B}^{U}_{i}\nsubseteq\operatorname{Core}(V,\mathcal{W}_{0})$.

It follows that

$\\{\mathcal{C}^{U,V}_{j}:U\in\mathcal{U},V\in\mathcal{V},j<m_{U,V}\\}\cup\\{\mathcal{B}^{U}_{i}:U\in\mathcal{U},i<n_{U}\\}$

is a partition of $\mathcal{W}_{0}$ in $\mathbin{E}$-connected sets, which
refines $\mathcal{U}$. Let $\mathcal{W}$ be the amalgam of $\mathcal{W}_{0}$
relative to this partition.

Given $W\in\mathcal{W}$ and $V\in\mathcal{V}$, by (5.2) we have that

(5.3) $W\in\operatorname{Core}(V,\mathcal{W})\iff
W=\bigvee\mathcal{C}^{U,V}_{j}\text{ for some }U\in\mathcal{U},j<m_{U,V}.$

This shows in particular that
$\operatorname{Core}(V,\mathcal{W})=\operatorname{Star}(T_{V},\mathcal{W})$,
which is nonempty and $E$-connected.

To verify item (2), pick $W\in\mathcal{W}$. If there is $V\in\mathcal{V}$ such
that $W\in\operatorname{Core}(V,\mathcal{W})$ then let $W^{\prime}\coloneqq
W$. If that is not the case, then by (5.3) we know that
$W=\bigvee\mathcal{B}^{U}_{i}$ for some $U\in\mathcal{U}$ and $i<n_{U}$. Since
$\mathcal{B}^{U}_{i}$ is a connected component of $\mathcal{B}^{U}$ and
$\mathcal{W}_{0}(U)$ is $E$-connected, there is some
$V\in\operatorname{Star}(U,\mathcal{V})$ and $j<m_{U,V}$ such that
$(\bigvee\mathcal{B}^{U}_{i})\mathbin{E}(\bigvee\mathcal{C}^{U,V}_{j})$. By
(5.3) we then have
$\bigvee\mathcal{C}^{U,V}_{j}\in\operatorname{Core}(V,\mathcal{W})$, as
desired. ∎

We recall that, under the identification of a partition $\mathcal{W}$ with its
nerve graph, the set $[W]_{1}$, for $W\in\mathcal{W}$, is the set of all
$W^{\prime}\in\mathcal{W}$ such that $W\mathbin{E}W^{\prime}$.

######  Theorem 5.6.

Let $X$ be a Peano continuum, $\mathcal{U},\mathcal{V}$ be brick partitions on
$X$, and let $h\in\mathrm{Homeo}(X)$. There are a brick partition
$\mathcal{W}\preceq\mathcal{U}$ and a monotone epimorphism
$\alpha\colon\mathcal{W}\to\mathcal{V}$ such that
$h[\operatorname{Cl}(W)]\subseteq\bigvee[\alpha(W)]_{1}$ for each
$W\in\mathcal{W}$.

Moreover, if $u,v$ are walks on $\mathcal{U},\mathcal{V}$, respectively, such
that $\mathbf{O}_{\mathcal{U},u}\subseteq
h^{-1}\cdot\mathbf{O}_{\mathcal{V},v}$, then $\mathcal{W}$ and $\alpha$ can be
chosen so that there exists a walk $w$ on $\mathcal{W}$ such that
$\rho^{\mathcal{W}}_{\mathcal{U}}\cdot w\sqsubseteq u$ and $\alpha\cdot
w\sqsubseteq v$.

###### Proof.

Let $\mathcal{W}$ be given by Proposition 5.5 applied to $\mathcal{U}$ and
$h^{-1}\cdot\mathcal{V}$. We claim that $\mathcal{W}$ can be picked fine
enough so that:

  1. (1)

if $W\mathbin{E}W^{\prime}$ then $V\mathbin{E}V^{\prime}$ for any
$V\in\operatorname{Star}(h[\operatorname{Cl}(W)],\mathcal{V}),V^{\prime}\in\operatorname{Star}(h[\operatorname{Cl}(W^{\prime})],\mathcal{V})$,

  2. (2)

for each $V\mathbin{E}V^{\prime}$ there is a path $P$ in
$\mathcal{W}(h^{-1}(V\vee V^{\prime}))$ connecting
$\operatorname{Core}(h^{-1}(V),\mathcal{W})$ to
$\operatorname{Core}(h^{-1}(V^{\prime}),\mathcal{W})$.

It is fairly clear how to satisfy the first condition, so we focus on the
second one: as $\mathcal{V}$ is a brick partition, given
$V,V^{\prime}\in\mathcal{V}$ with $V\mathbin{E}V^{\prime}$, by Facts 4.2 and
4.3 we can find an arc $\gamma\in O(V,V^{\prime})$. By choosing $\mathcal{W}$
fine enough we can assume that
$\operatorname{Star}(h^{-1}(\gamma),\mathcal{W})\subseteq\mathcal{W}(h^{-1}(V\vee
V^{\prime}))$ and that there are
$W\in\operatorname{Core}(h^{-1}(V),\mathcal{W})\cap\operatorname{Star}(h^{-1}(\gamma),\mathcal{W})$
and
$W^{\prime}\in\operatorname{Core}(h^{-1}(V^{\prime}),\mathcal{W})\cap\operatorname{Star}(h^{-1}(\gamma),\mathcal{W})$.
Now let $P$ be any path in $\operatorname{Star}(h^{-1}(\gamma),\mathcal{W})$
from $W$ to $W^{\prime}$. Since this can be done for any pair of $E$-adjacent
elements in $\mathcal{V}$, the conclusion follows.

We define $\alpha$ in two steps. First, for each $V\in\mathcal{V}$ and
$W\in\operatorname{Core}(h^{-1}(V),\mathcal{W})$, let $\alpha(W)\coloneqq V$.
Then, by Proposition 5.5, for every $W\in\mathcal{W}$ there is at least one
$W^{\prime}\mathbin{E}W$ on which $\alpha$ is defined. Pick one such
$W^{\prime}$ and let $\alpha(W)\coloneqq\alpha(W^{\prime})$. Notice that in
particular $W\mathbin{E}\mathcal{W}(h^{-1}(\alpha(W)))$, so
$h[\operatorname{Cl}(W)]\cap\operatorname{Cl}(\alpha(W))\neq\emptyset$, that
is:

(5.4) $\alpha(W)\in\operatorname{Star}(h[\operatorname{Cl}(W)],\mathcal{V}).$

Moreover, if $W\in\mathcal{W}(\bigvee_{V\in\mathcal{V}_{0}}h^{-1}(V))$ for
some $\mathcal{V}_{0}\subseteq\mathcal{V}$, then

(5.5) $\alpha(W)\in\mathcal{V}_{0},$

since if $W^{\prime}\mathbin{E}W$ and
$W^{\prime}\in\operatorname{Core}(h^{-1}(V),\mathcal{W})$, then
$V\in\mathcal{V}_{0}$.

Condition (1) and (5.4) imply that $\alpha$ is a graph homomorphism, while
condition (2) and (5.5) imply that $\alpha$ is surjective on edges.

For each $V\in\mathcal{V}$, $\alpha^{-1}(V)$ contains
$\mathcal{W}(h^{-1}(V))$, which is nonempty and $\mathbin{E}$-connected by
Proposition 5.5.(1), and by definition of $\alpha$ any other element of
$\alpha^{-1}(V)$ is adjacent to $\mathcal{W}(h^{-1}(V))$, so $\alpha^{-1}(V)$
is $\mathbin{E}$-connected.

Notice that condition (1) implies that for each $W\in\mathcal{W}$,
$\operatorname{Star}(h[\operatorname{Cl}(W)],\mathcal{V})$ is a clique in
$\mathcal{V}$. Together with (5.4), it follows that

$h[\operatorname{Cl}(W)]\subseteq\bigvee\operatorname{Star}(h[\operatorname{Cl}(W)],\mathcal{V})\subseteq\bigvee[\alpha(W)]_{1}\text{
for every }W\in\mathcal{W}.$

For the moreover, fix $\mathcal{C}\in\mathbf{O}_{\mathcal{U},u}\subseteq
h^{-1}\cdot\mathbf{O}_{\mathcal{V},v}$. For each $s<\operatorname{lh}(u)$, let
$C_{s}^{u}\in\mathcal{C}$ be such that $C_{s}^{u}\in O(u(0),\dots,u(s))$, and
for each $t<\operatorname{lh}(v)$, let $C_{t}^{v}\in\mathcal{C}$ be such that
$C_{t}^{v}\in O(h^{-1}(v(0)),\dots,h^{-1}(v(t)))$. We can suppose that
$C_{0}^{u}=C_{0}^{v}$ is the root $\bigcap\mathcal{C}$ of $\mathcal{C}$.

Pick $\mathcal{W}$ which is moreover fine enough that for each
$t<\operatorname{lh}(v)$:

  1. (a)

$\operatorname{Star}(C_{t}^{v},\mathcal{W})\subseteq\mathcal{W}(\bigvee_{j\leq
t}h^{-1}(v(j)))$,

  2. (b)

there is
$W_{t}\in\operatorname{Star}(C_{t}^{v},\mathcal{W})\cap\mathcal{W}(h^{-1}(v(t)))$.

Let $C_{0}\subsetneq C_{1}\subsetneq\cdots\subsetneq C_{n-1}$ enumerate
$\\{C_{s}^{u}\\}_{s<\operatorname{lh}(u)}\cup\\{C_{t}^{v}\\}_{t<\operatorname{lh}(v)}$
without repetitions, and let $w$ be any walk which starts in $W_{0}$ and
covers all of $\operatorname{Star}(C_{k},\mathcal{W})$ before passing to
$\operatorname{Star}(C_{k+1},\mathcal{W})$. Call $k_{t}<\operatorname{lh}(w)$
the minimum such that
$\operatorname{Star}(C_{t}^{v},\mathcal{W})=\\{w(0),\dots,w(k_{t})\\}$.

Since $\mathcal{W}$ refines $\mathcal{U}$,

$O(\operatorname{Star}(C_{s}^{u},\mathcal{W}))\subseteq O(u(0),\dots,u(s)),$

for all $s<\operatorname{lh}(u)$, so $\rho^{\mathcal{W}}_{\mathcal{U}}\cdot
w\sqsubseteq u$. Fix $t<\operatorname{lh}(v)$. By (a) and (5.5),

$\alpha[\\{w(0),\dots,w(k_{t})\\}]=\alpha[\operatorname{Star}(C_{t}^{v},\mathcal{W})]\subseteq\\{(v(0)),\dots,(v(t))\\}.$

On the other hand, for each $j\leq t$, $W_{j}\in\\{w(0),\dots,w(k_{t})\\}$,
and by (b) and the definition of $\alpha$, $\alpha(W_{j})=v(j)$, so

$\alpha[\\{w(0),\dots,w(k_{t})\\}]=\\{(v(0)),\dots,(v(t))\\}.$

Since $t$ was arbitrary, $\alpha\cdot w\sqsubseteq v$. ∎

Each monotone epimorphism $\alpha\colon\mathcal{W}\to\mathcal{V}$ between
brick partitions of $X$ gives rise to an open subset of $\mathrm{Homeo}(X)$:

$H_{\alpha}\coloneqq\\{h\in\mathrm{Homeo}(X):h[\operatorname{Cl}(W)]\subseteq\bigvee[\alpha(W)]_{1},\text{
for each }W\in\mathcal{W}\\}.$

While many such $H_{\alpha}$’s might be empty, Theorem 5.6 implies that their
collection forms a basis for the compact-open topology on $\mathrm{Homeo}(X)$.
In particular the open sets

(5.6) $H_{\mathcal{V},1}\coloneqq
H_{\rho^{\mathcal{V}}_{\mathcal{V}}}=\\{h\in\mathrm{Homeo}(X):h[\operatorname{Cl}(V)]\subseteq\bigvee[V]_{1},\text{
for each }V\in\mathcal{V}\\},$

form a basis of open neighborhood of the identity in $\mathrm{Homeo}(X)$, when
$\mathcal{V}$ varies among brick partitions of $X$.

We are finally ready for the main result of this section.

######  Theorem 5.7.

Let $X$ be a Peano continuum with a generic chain. Then for each brick
partition $\mathcal{U}$ of $X$ and walk $u$ on $\mathcal{U}$ there are a brick
partition and a walk $(\mathcal{V},v)\preceq(\mathcal{U},u)$ such that for any
brick partition $\mathcal{W}\preceq\mathcal{V}$ and any two walks
$w_{0},w_{1}$ on $\mathcal{W}$ with $\rho^{\mathcal{W}}_{\mathcal{V}}\cdot
w_{i}\sqsubseteq v$, for $i=0,1$, there are a brick partition $\mathcal{Z}$, a
walk $z$ on $\mathcal{Z}$ and two monotone epimorphisms
$\alpha_{0},\alpha_{1}\colon\mathcal{Z}\to\mathcal{W}$ satisfying:

  1. (1)

$\alpha_{i}\cdot z\sqsubseteq w_{i}$, for $i=0,1$;

  2. (2)

$d_{\sup}(\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{0},\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{1})\leq
1$.

###### Proof.

Let $\mathcal{U}$ and a walk $u$ on $\mathcal{U}$ be given. Let
$(\mathcal{U}_{0},u_{0})\preceq(\mathcal{U},u)$, with $\mathcal{U}_{0}$ a
brick partition fine enough that

$d_{\mathcal{U}}(\rho^{\mathcal{U}_{0}}_{\mathcal{U}}(U),\rho^{\mathcal{U}_{0}}_{\mathcal{U}}(U^{\prime}))\leq
1\text{ whenever }d_{\mathcal{U}_{0}}(U,U^{\prime})\leq 2.$

Find a symmetric open neighborhood of the identity $H_{0}\subseteq
H_{\mathcal{U}_{0},1}$ (see (5.6)). Also, set
$\mathbf{U}\coloneqq\mathbf{O}_{\mathcal{U}_{0},u_{0}}$. Theorem 5.4 gives us
a nonempty $\mathbf{V}\subseteq\mathbf{U}$. By Proposition 5.2, there are
$(\mathcal{V},v)\preceq(\mathcal{U}_{0},u_{0})\preceq(\mathcal{U},u)$, with
$\mathcal{V}$ a brick partition, such that
$\mathbf{O}_{\mathcal{V},v}\subseteq\mathbf{V}$.

Suppose now that we are given a brick partition
$\mathcal{W}\preceq\mathcal{V}$ and two walks $w_{0},w_{1}$ on $\mathcal{W}$
with $\rho^{\mathcal{W}}_{\mathcal{V}}\cdot w_{i}\sqsubseteq v$, for $i=0,1$.
Set $\mathbf{W}_{i}\coloneqq\mathbf{O}_{\mathcal{W},w_{i}}$; then
$\mathbf{W}_{i}\subseteq\mathbf{O}_{\mathcal{V},v}\subseteq\mathbf{V}$, for
$i=0,1$. Therefore by Theorem 5.4 there is $h\in H_{0}$ such that
$h^{-1}\cdot\mathbf{W}_{0}\cap\mathbf{W}_{1}$ is a nonempty open set. We can
thus apply Proposition 5.2 to obtain
$(\mathcal{Z}^{\prime},z^{\prime})\preceq(\mathcal{W},w_{1})$ such that
$\mathbf{O}_{\mathcal{Z}^{\prime},z^{\prime}}\subseteq
h^{-1}\cdot\mathbf{W}_{0}\cap\mathbf{W}_{1}\subseteq\mathbf{W}_{1}$.

By Theorem 5.6, there are a brick partition
$\mathcal{Z}\preceq\mathcal{Z}^{\prime}$, a walk $z$ on $\mathcal{Z}$ and a
monotone epimorphism $\alpha_{0}\colon\mathcal{Z}\to\mathcal{W}$ such that
$\rho^{\mathcal{Z}}_{\mathcal{Z}^{\prime}}\cdot z\sqsubseteq z^{\prime}$,
$\alpha_{0}\cdot z\sqsubseteq w_{0}$ and such that, for each
$Z\in\mathcal{Z}$,

(5.7) $h[\operatorname{Cl}(Z)]\subseteq\bigvee[\alpha_{0}(Z)]_{1}.$

Call $\alpha_{1}\coloneqq\rho^{\mathcal{Z}}_{\mathcal{W}}$; then
$\alpha_{1}\cdot z\sqsubseteq w_{1}$.

It remains to prove item (2). Fix $Z\in\mathcal{Z}$. On the one hand,

$h[\operatorname{Cl}(Z)]\subseteq
h[\operatorname{Cl}(\rho^{\mathcal{Z}}_{\mathcal{U}_{0}}(Z))]\subseteq\bigvee[\rho^{\mathcal{Z}}_{\mathcal{U}_{0}}(Z)]_{1},$

since $h\in H_{0}\subseteq H_{\mathcal{U}_{0},1}$. On the other hand,

$h[\operatorname{Cl}(Z)]\subseteq\bigvee[\alpha_{0}(Z)]_{1}\subseteq\bigvee[\rho^{\mathcal{W}}_{\mathcal{U}_{0}}\alpha_{0}(Z)]_{1},$

by (5.7). Therefore,
$\bigvee[\rho^{\mathcal{W}}_{\mathcal{U}_{0}}\alpha_{0}(Z)]_{1}\cap\bigvee[\rho^{\mathcal{Z}}_{\mathcal{U}_{0}}(Z)]_{1}\neq\emptyset$,
so

$d_{\mathcal{U}_{0}}(\rho^{\mathcal{W}}_{\mathcal{U}_{0}}\alpha_{0}(Z),\rho^{\mathcal{Z}}_{\mathcal{U}_{0}}(Z))\leq
2.$

It follows from (5.1) that
$d_{\mathcal{U}}(\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{0}(Z),\rho^{\mathcal{Z}}_{\mathcal{U}}(Z))\leq
1$. Since $Z\in\mathcal{Z}$ was arbitrary,
$d_{\sup}(\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{0},\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{1})=d_{\sup}(\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{0},\rho^{\mathcal{Z}}_{\mathcal{U}})\leq
1$, as wished. ∎

##  6\. Absence of Generic Chains

In this section we finally put all pieces together and prove Theorem 1.2. The
first step is to show that the hypothesis that $X$ has no locally separating
points allows us to only deal with walks which are spaced paths. We do so in
Theorem 6.2 ( (Upgrading Walks to Paths).). This will be helpful when trying
to use Lemma 3.3 and Lemma 3.4 to compare winding numbers of walks defined on
different partitions.

To do so, we need the following proposition, which is a combinatorial version
of Arc Doubling (see [MOT]*Lemma 3.4).

######  Proposition 6.1 (Path Doubling).

Let $X$ be a Peano continuum with no locally separating points. Let
$\mathcal{U}$ be a brick partition of $X$ and $u$ be a path on $\mathcal{U}$.
There are a brick partition $\mathcal{V}\preceq\mathcal{U}$ and $2$ paths
$v_{0},v_{1}$ on $\mathcal{V}$ which are disjoint except for their endpoints
and such that $\rho^{\mathcal{V}}_{\mathcal{U}}\cdot v_{i}$ monotonically
refines $u$ for $i=0,1$.

###### Proof.

For each $i<\operatorname{lh}(u)-1$, let $A_{i}$ be a subset of cardinality 2
of $\operatorname{Cl}(u(i))\cap\operatorname{Cl}(u(i+1))\cap(u(i)\vee
u(i+1))$, which exists by 4.2.(2). Given $1<i<\operatorname{lh}(u)-1$, 4.3.(2)
gives two disjoint arcs $\gamma_{i},\gamma^{\prime}_{i}$ in $u(i)\cup
A_{i}\cup A_{i+1}$ connecting $A_{i}$ to $A_{i+1}$. Since $u$ is a path, all
the $A_{i}$’s and all the $\gamma_{i}$’s are distinct, and the $\gamma_{i}$’s
intersect at most in their endpoints. The same holds for the
$\gamma^{\prime}_{i}$’s. Let $\lambda,\lambda^{\prime}$ be the two disjoint
arcs resulting from the concatenations of the $\gamma_{i}$’s and
$\gamma^{\prime}_{i}$’s respectively.

Let $\tilde{\mathcal{V}}\preceq\mathcal{U}$ be a brick partition fine enough
that
$\operatorname{Star}(\lambda,\tilde{\mathcal{V}})\cap\operatorname{Star}(\lambda^{\prime},\tilde{\mathcal{V}})=\emptyset$.
Let, for each $1<i<\operatorname{lh}(u)-1$:

$\displaystyle P_{i}$
$\displaystyle\coloneqq\operatorname{Star}(\lambda,\tilde{\mathcal{V}}(u(i))),$
$\displaystyle P^{\prime}_{i}$
$\displaystyle\coloneqq\operatorname{Star}(\lambda^{\prime},\tilde{\mathcal{V}}(u(i))),$
$\displaystyle P_{0}=P^{\prime}_{0}$
$\displaystyle\coloneqq\tilde{\mathcal{V}}(u(0)),$ $\displaystyle
P_{\operatorname{lh}(u)-1}=P^{\prime}_{\operatorname{lh}(u)-1}$
$\displaystyle\coloneqq\tilde{\mathcal{V}}(u(-1)).$

Let $\mathcal{V}$ be an amalgam of $\tilde{\mathcal{V}}$ refining
$\mathcal{U}$ and containing each $\bigvee P_{i}$ and each $\bigvee
P^{\prime}_{i}$, for $i<\operatorname{lh}(u)$. Then $v_{0}\coloneqq\langle
P_{0},\dots,P_{\operatorname{lh}(u)-1}\rangle$ and $v_{1}\coloneqq\langle
P^{\prime}_{0},\dots,P^{\prime}_{\operatorname{lh}(u)-1}\rangle$ are paths on
$\mathcal{V}$ satisfying the conclusions of the proposition. ∎

######  Theorem 6.2 (Upgrading Walks to Paths).

Let $X$ be a Peano continuum with no locally separating points. Let
$\mathcal{V}$ be a brick partition of $X$ and $v$ be a reduced walk on
$\mathcal{V}$. There is a spaced path $w$ on a brick partition $\mathcal{W}$
such that $(\mathcal{W},w)\preceq(\mathcal{V},v)$. If furthermore $v$ is
uncrossed then $w$ can be chosen so that
$\rho^{\mathcal{W}}_{\mathcal{V}}\cdot w$ monotonically refines $v$.

###### Proof.

We proceed by induction on the length $\ell$ of $v$. If $\ell=1$ let
$\mathcal{W}\coloneqq\mathcal{V}$ and $w\coloneqq v$. Suppose the result is
true for $\ell\geq 1$ and let us prove it for $v=\langle
v(0),\dots,v(\ell)\rangle$. By the induction hypothesis, there is a brick
partition $\mathcal{Z}\preceq\mathcal{V}$ and a spaced path $z$ on
$\mathcal{Z}$ such that $\rho^{\mathcal{Z}}_{\mathcal{V}}\cdot z$ refines
$\langle v(0),\dots,v(\ell-1)\rangle$, and it does so monotonically if the
latter is uncrossed.

By  (Path Doubling). (Proposition 6.1), there are a brick partition
$\mathcal{W}\preceq\mathcal{Z}$ and spaced paths $w_{0},w_{1}$ on
$\mathcal{W}$, which are disjoint except for their endpoints, and such that
$\rho^{\mathcal{W}}_{\mathcal{Z}}\cdot w_{i}$ monotonically refines $z$, for
$i<2$. Choose any $W\in\mathcal{W}(v(\ell))$ with
$W\mathbin{E}\mathcal{W}(v(\ell-1))$ and let $\bar{w}$ be a path from
$w_{0}(-1)=w_{1}(-1)$ to $W$ in $\mathcal{W}(v(\ell-1))\cup\\{W\\}$.

If $v$ is uncrossed: Suppose without loss of generality that the _last_ time
that $\bar{w}$ intersects the $\mathbin{E}$-connected component of $(w_{0}\cup
w_{1})\cap\mathcal{W}(v(\ell))$ containing $w_{0}(-1)=w_{1}(-1)$ is in
$w_{0}(i)$, for some $i<\operatorname{lh}(w_{0})$. If after that step
$\bar{w}$ is disjoint from $(w_{0}\cup w_{1})$, then we can simply take $w$ to
be the concatenation of $w_{0}$ up to $w_{0}(i)$ and of the final segment of
$\bar{w}$. Otherwise, let $w_{\epsilon}(j)$ be the _first_ time $\bar{w}$
intersects $(w_{0}\cup w_{1})$ after $w_{0}(i)$.

If $\epsilon=0$, let $w$ be the walk that follows $w_{1}$ from start to end,
then follows $w_{0}$ backwards from $w_{0}(-1)$ to $w_{0}(i)$, then $\bar{w}$
until $w_{0}(j)$, then $w_{0}$ until it exits $\mathcal{W}(v(\ell-1))$ and
crosses into $\mathcal{W}(v(\ell))$, which is the case since $v$ is uncrossed
and $\rho^{\mathcal{W}}_{\mathcal{V}}\cdot w_{0}$ monotonically refines
$\langle v(0),\dots,v(\ell-1)\rangle$.

If $\epsilon=1$, let $w$ be the walk that follows $w_{0}$ from start to
$w_{0}(i)$, then $\bar{w}$ until $w_{1}(j)$, then $w_{1}$ until it exits
$\mathcal{W}(v(\ell-1))$ and crosses into $\mathcal{W}(v(\ell))$, which is the
case since $v$ is uncrossed and $\rho^{\mathcal{W}}_{\mathcal{V}}\cdot w_{1}$
monotonically refines $\langle v(0),\dots,v(\ell-1)\rangle$. In either case,
$\rho^{\mathcal{W}}_{\mathcal{V}}\cdot w$ monotonically refines $v$.

Otherwise, we have to content ourselves to refine non-monotonically. Suppose
without loss of generality that the _last_ time that $\bar{w}$ intersects
$w_{0}\cup w_{1}$ is in $w_{0}(j)$, for some $j<\operatorname{lh}(w_{0})$. Let
$w$ be the walk that follows $w_{1}$ from start to end, then follows $w_{0}$
backwards from $w_{0}(-1)$ to $w_{0}(j)$ (this part might exit and then re-
enter $\mathcal{W}(v(\ell-1))$) and then $\bar{w}$ from $w_{0}(j)$ to $W$.

By  (Spacing out Paths). (Lemma 5.3), we can further ensure that $w$ is
spaced. ∎

###  6.1. Robust cycles

The next step is to find a sufficient condition on $X$ under which the
conclusion of Theorem 5.7, a necessary consequence to the existence of a
generic chain on $X$, is violated.

Broadly speaking, our goal is to isolate a region of $X$ (a circular
subpartition to be precise) inside which there is well-defined notion of
winding number, while also making sure that we can refine and extend given
paths in such region so as to be able to reach arbitrarily large positive and
negative values of winding number. Once this is ensured, if $(\mathcal{V},v)$
is as in the statement of Theorem 5.7, we can then find a brick partition
$\mathcal{W}\preceq\mathcal{V}$ and two paths $w_{0},w_{1}$ on $\mathcal{W}$
refining $v$ and such that the winding number of $w_{0}$ is very large, while
the one of $w_{1}$ is negative and very large in absolute value. If $X$ had a
generic chain, Theorem 5.7 would permit to amalgamate up to distance one
$(\mathcal{W},w_{0})$ and $(\mathcal{W},w_{1})$, and this would lead to a
contradiction by Lemma 3.5, since $w_{0}$ and $w_{1}$ were chosen to have very
distant winding numbers. The idea of using the winding number to show the
failure of such amalgamation goes back to the proof of [GTZ]*Theorem 1.2.

[scale=1]img/robust

Figure 1. A robust cycle.

Once this is done, all that is left is to show that the spaces considered in
Theorem 1.2 satisfy the necessary property to perform this construction (this
is done in Theorems 6.6, 6.10 and 6.12). Such property is formalized in
Definition 6.3 below. The intuitive and simplified idea is to require the
existence of a partition $\mathcal{S}$ of $X$ with a _robust_ cycle $C$: by
this we mean a cycle such that, given any brick partition
$\mathcal{V}\preceq\mathcal{S}$ and walk $v$ on $\mathcal{V}$, even in the
situation where $v$ intersects and apparently _cuts_ $C$, it is always
possible to _zoom in_ (that is, find a refinement) to see that $C$ is still
intact and in fact $v$ can be prolonged with a circular path confined in $C$
in a strong sense.

######  Definition 6.3.

Let us say that a Peano continuum $X$ has a _robust cycle_ if there are a
brick partition $\mathcal{S}$ with a spaced circular path $C$ of length
$\ell\geq 4$, and a brick partition $\mathcal{U}\preceq\mathcal{S}$ with a
walk $u$ on $\mathcal{U}$ such that for any brick partition and walk
$(\mathcal{V},v)\preceq(\mathcal{U},u)$ there are
$(\mathcal{V}^{\prime},v^{\prime}{}^{\frown}c)\preceq(\mathcal{V},v)$ where
$v^{\prime}{}^{\frown}c$ is a lasso path such that:

(6.1)
$\rho^{\mathcal{U}}_{\mathcal{S}}([\rho^{\mathcal{V}^{\prime}}_{\mathcal{U}}\cdot
c]_{2})\subseteq C,$

and

(6.2)
$\lvert\operatorname{wd}_{C}(\rho^{\mathcal{V}^{\prime}}_{\mathcal{S}}\cdot
c)\rvert\geq\ell.$

######  Theorem 6.4.

Let $X$ be a Peano continuum with no locally separating points. If $X$ has a
robust cycle, then $X$ has no generic chain.

###### Proof.

The existence of a robust cycle gives us a brick partition $\mathcal{S}$, a
spaced circular path $C$ of length $\ell\geq 4$, a brick partition
$\mathcal{U}\preceq\mathcal{S}$, and and a walk $u$ on $\mathcal{U}$. Up to
taking a refinement, by Lemma 3.1 we can suppose that

(6.3)
$d_{\mathcal{S}}(\rho^{\mathcal{U}}_{\mathcal{S}}(U),\rho^{\mathcal{U}}_{\mathcal{S}}(U^{\prime}))\leq
1\text{ whenever }d_{\mathcal{U}}(U,U^{\prime})\leq 2.$

Suppose we are given a brick partition and a walk
$(\mathcal{V},v_{0})\preceq(\mathcal{U},u)$. By definition of robust cycle and
by  (Spacing out Paths). (Lemma 5.3) we can assume, up to a refinement, that
$v_{0}$ is a spaced lasso path $v{}^{\frown}c_{+}$ such that:

(6.4) $\rho^{\mathcal{U}}_{\mathcal{S}}([\rho^{\mathcal{V}}_{\mathcal{U}}\cdot
c_{+}]_{2})\subseteq C,$

and

$\lvert\operatorname{wd}_{C}(\rho^{\mathcal{V}}_{\mathcal{S}}\cdot
c_{+})\rvert\geq\ell.$

Let $c_{-}$ be the spaced circular path going in the opposite direction of
$c_{+}$. Set
$N\coloneqq\operatorname{wd}_{C}(\rho^{\mathcal{V}}_{\mathcal{S}}\cdot
c_{+})$, so that $\operatorname{wd}_{C}(\rho^{\mathcal{V}}_{\mathcal{S}}\cdot
c_{-})=-N$. Up to switching the role of $c_{+},c_{-}$, we might assume that
$N\geq\ell>0$.

Let

$M_{c}\coloneqq\max\\{\lvert\operatorname{wd}_{C}(\rho^{\mathcal{V}}_{\mathcal{S}}\cdot
t)\rvert:t\text{ is an initial segment of }c_{+}\text{ or }c_{-}\\},$

and

$M_{v}\coloneqq\max\\{\lvert\operatorname{wd}_{C}(\rho^{\mathcal{V}}_{\mathcal{S}}\cdot
t)\rvert:t\text{ is an initial or final segment of }v\\}.$

Since $N\geq\ell>1$, there is $k\in\mathbb{N}$ such that

$k\cdot N>3M_{v}+M_{c}+2.$

Let

$\bar{w}_{0}\coloneqq\underbrace{c_{+}{}^{\frown}c_{+}{}^{\frown}\cdots{}^{\frown}c_{+}}_{k\text{
times}}\quad\text{and}\quad\bar{w}_{1}\coloneqq\underbrace{c_{-}{}^{\frown}c_{-}{}^{\frown}\cdots{}^{\frown}c_{-}}_{k\text{
times}}.$

Since $v{}^{\frown}\bar{w}_{0}$ and $v{}^{\frown}\bar{w}_{1}$ are uncrossed,
by  (Upgrading Walks to Paths). (Theorem 6.2) twice, we find a brick partition
$\mathcal{W}\preceq\mathcal{V}$ and spaced paths $w_{0},w_{1}$ on
$\mathcal{W}$ such that $\rho^{\mathcal{W}}_{\mathcal{V}}\cdot w_{i}$
_monotonically_ refines $v{}^{\frown}\bar{w}_{i}$. We stress that
$w_{0},w_{1}$ are paths, _not_ lassos.

Using Lemma 3.3 and the definitions of $w_{0},w_{1}$, we obtain the following
computations of winding numbers:

######  Claim 6.4.1.

  1. (a0)

If $t$ is an initial segment of $w_{0}$ then:

$\operatorname{wd}_{C}(\rho^{\mathcal{W}}_{\mathcal{S}}\cdot
t)\geq-(M_{v}+M_{c}),$

  2. (a1)

If $t$ is an initial segment of $w_{1}$ then:

$\operatorname{wd}_{C}(\rho^{\mathcal{W}}_{\mathcal{S}}\cdot t)\leq
M_{v}+M_{c},$

  3. (b0)

If $t$ is a final segment of $w_{0}$ such that
$\rho^{\mathcal{W}}_{\mathcal{V}}\cdot t$ contains $\bar{w}_{0}$ then:

$\operatorname{wd}_{C}(\rho^{\mathcal{W}}_{\mathcal{S}}\cdot
t)>2M_{v}+M_{c}+2,$

  4. (b1)

If $t$ is a final segment of $w_{1}$ such that
$\rho^{\mathcal{W}}_{\mathcal{V}}\cdot t$ contains $\bar{w}_{1}$ then:

$\operatorname{wd}_{C}(\rho^{\mathcal{W}}_{\mathcal{S}}\cdot
t)<-(2M_{v}+M_{c}+2).$

We now apply Theorem 5.7, hence towards a contradiction suppose there are a
brick partition $\mathcal{Z}$, two monotone epimorphisms
$\alpha_{0},\alpha_{1}\colon\mathcal{Z}\to\mathcal{W}$, and a reduced walk $z$
on $\mathcal{Z}$ such that $\alpha_{i}\cdot z\sqsubseteq w_{i}$, for $i=0,1$,
and

(6.5)
$d_{\sup}(\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{0},\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{1})\leq
1.$

Let $\bar{z}$ be the shortest initial segment of $z$ such that there is
$i\in\\{0,1\\}$ for which $\alpha_{i}\cdot\bar{z}\sqsubseteq w_{i}$.
Necessarily, $\alpha_{i}\cdot\bar{z}$ is confined in $w_{i}$. Let us suppose
$i=0$, the case $i=1$ being symmetrical.

We will find a contradiction by examining
$\operatorname{wd}_{C}(\rho^{\mathcal{W}}_{\mathcal{S}}\alpha_{1}\cdot\bar{z})$.
On one hand $\alpha_{1}\cdot\bar{z}$ refines and is confined to an initial
segment of the spaced path $w_{1}$, so by 6.4.1.(a1) and Lemma 3.4

(6.6)
$\operatorname{wd}_{C}(\rho^{\mathcal{W}}_{\mathcal{S}}\alpha_{1}\cdot\bar{z})\leq
M_{v}+M_{c}.$

On the other hand, write $\bar{z}$ as $\bar{z}_{v}{}^{\frown}\bar{z}_{0}$,
where $\bar{z}_{0}$ is the longest final segment such that

(6.7)
$\rho^{\mathcal{U}}_{\mathcal{S}}([\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{0}\cdot\bar{z}_{0}]_{1})\subseteq
C.$

Notice that $\bar{z}_{v}$ might be the empty walk (this would be the case if
$C=\mathcal{S}$). In particular, $\alpha_{0}\cdot\bar{z}_{0}$ refines and is
confined to a final segment of $w_{0}$ whose image under
$\rho^{\mathcal{W}}_{\mathcal{V}}$ contains $\bar{w}_{0}$, and moreover
$\alpha_{0}\cdot\bar{z}_{0}(-1)=w_{0}(-1)$, so by 6.4.1.(b0) and Lemma 3.4

(6.8)
$\operatorname{wd}_{C}(\rho^{\mathcal{W}}_{\mathcal{S}}\alpha_{0}\cdot\bar{z}_{0})>2M_{v}+M_{c}+2.$

By (6.5) and (6.7) we know that
$\rho^{\mathcal{W}}_{\mathcal{S}}\alpha_{1}\cdot\bar{z}_{0}$ is confined to
$C$. We moreover have that

$d_{\mathcal{U}}(\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{0}\cdot\bar{z}_{0}(i),\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{1}\cdot\bar{z}_{0}(i))\leq
1\text{ for each }i<\operatorname{lh}(\bar{z}_{0}),$

and

$d_{\mathcal{U}}(\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{0}\cdot\bar{z}_{0}(j),\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{1}\cdot\bar{z}_{0}(j+1))\leq\\\
\leq
d_{\mathcal{U}}(\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{0}\cdot\bar{z}_{0}(j),\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{0}\cdot\bar{z}_{0}(j+1))+d_{\mathcal{U}}(\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{0}\cdot\bar{z}_{0}(j+1),\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{1}\cdot\bar{z}_{0}(j+1))\leq
2,$

for each $j<\operatorname{lh}(\bar{z}_{0})-1$. Using (6.3) we get
$d_{\mathcal{S}}(\rho^{\mathcal{W}}_{\mathcal{S}}\alpha_{0}\cdot\bar{z}_{0}(j),\rho^{\mathcal{W}}_{\mathcal{S}}\alpha_{1}\cdot\bar{z}_{0}(j+1))\leq
1$, so we can apply Lemma 3.5 and (6.8) to conclude

(6.9)
$\operatorname{wd}_{C}(\rho^{\mathcal{W}}_{\mathcal{S}}\alpha_{1}\cdot\bar{z}_{0})>2M_{v}+M_{c}.$

If $\bar{z}_{v}$ is the empty walk then
$\operatorname{wd}_{C}(\rho^{\mathcal{W}}_{\mathcal{S}}\alpha_{1}\cdot\bar{z})=\operatorname{wd}_{C}(\rho^{\mathcal{W}}_{\mathcal{S}}\alpha_{1}\cdot\bar{z}_{0})$,
so we obtain a contradiction from (6.6) and (6.9). Otherwise
$\alpha_{1}\cdot\bar{z}_{v}$ refines an initial segment of $w_{1}$ and
$\rho^{\mathcal{W}}_{\mathcal{V}}\alpha_{1}(\bar{z}_{v}(-1))$ is contained in
$v$, and not in $\bar{w}_{1}$: indeed by definition of $\bar{z}_{0}$ the
vertex $\rho^{\mathcal{W}}_{\mathcal{U}}(\alpha_{0}(\bar{z}_{v}(-1)))$ must be
$E$-adjacent to one in $(\rho^{\mathcal{U}}_{\mathcal{S}})^{-1}(C)$, and this
in combination with (6.5) yields

$d_{\mathcal{U}}(\rho^{\mathcal{W}}_{\mathcal{U}}\alpha_{1}(\bar{z}_{v}(-1)),\mathcal{U}\setminus(\rho^{\mathcal{U}}_{\mathcal{S}})^{-1}(C))\leq
2,$

whereas
$\rho^{\mathcal{U}}_{\mathcal{S}}([\rho^{\mathcal{V}}_{\mathcal{U}}\cdot\bar{w}_{1}]_{2})=\rho^{\mathcal{U}}_{\mathcal{S}}([\rho^{\mathcal{V}}_{\mathcal{U}}\cdot
c_{+}]_{2})\subseteq C$, by (6.4). Lemma 3.4 and the definition of $M_{v}$
then give

(6.10)
$\lvert\operatorname{wd}_{C}(\rho^{\mathcal{W}}_{\mathcal{S}}\alpha_{1}\cdot\bar{z}_{v})\rvert\leq
M_{v}.$

Combining (6.9) and (6.10) we conclude
$\operatorname{wd}_{C}(\rho^{\mathcal{W}}_{\mathcal{S}}\alpha_{2}\cdot\bar{z})>M_{v}+M_{c}$,
again a contradiction with (6.6). ∎

In the next three subsections we verify that each of the three conditions
considered in Theorem 1.2 is enough to guarantee the existence of a robust
cycle, and therefore is covered by Theorem 6.4.

###  6.2. Locally non-planar open subset

The next proposition is a combinatorial translation of the Disjoint Arcs
Property (see [MR1373283]*Theorem 1).

######  Proposition 6.5 (Disjoint Path Property).

Let $X$ be a Peano continuum with no locally separating points, and let
$O\subseteq X$ be a locally non-planar open set. Let $\mathcal{U}$ be a brick
partition of $X$ and $u,u^{\prime}$ be paths on $\mathcal{U}$, such that $u$
is confined to $\mathcal{U}(O)$. There are a brick partition
$\mathcal{V}\preceq\mathcal{U}$ and paths $v,v^{\prime}$ on $\mathcal{V}$
which are disjoint and such that $\rho^{\mathcal{V}}_{\mathcal{U}}\cdot
v,\rho^{\mathcal{V}}_{\mathcal{U}}\cdot v^{\prime}$ monotonically refine
$u,u^{\prime}$, respectively. If $u$ is circular, $v$ can be chosen to be
circular.

###### Proof.

For each $1<i<\operatorname{lh}(u)$, let $x_{i}$ be a point in
$\operatorname{Cl}(u(i-1))\cap\operatorname{Cl}(u(i))\cap(u(i-1)\vee u(i))$,
and, for each $1<j<\operatorname{lh}(u^{\prime})$, let $x^{\prime}_{j}$ be a
point in
$\operatorname{Cl}(u^{\prime}(j-1))\cap\operatorname{Cl}(u^{\prime}(j))\cap(u^{\prime}(j-1)\vee
u^{\prime}(j))$. We can assume that all these points are distinct by 4.2.(2),
since $u$ and $u^{\prime}$ are paths, so for each $i<\operatorname{lh}(u)$
there is at most one $j<\operatorname{lh}(u^{\prime})$ such that
$u(i)=u^{\prime}(j)$, and vice versa.

If $u(i)\not=u^{\prime}(j)$ for every $j<\operatorname{lh}(u^{\prime})$, let
$\gamma_{i}$ be any arc in $u(i)\cup\\{x_{i},x_{i+1}\\}$ connecting $x_{i}$ to
$x_{i+1}$. Similarly, if for $j<\operatorname{lh}(u^{\prime})$ there is no $i$
such that $u^{\prime}(j)=u(i)$, let $\gamma^{\prime}_{j}$ be any arc in
$u^{\prime}(j)\cup\\{x^{\prime}_{j},x^{\prime}_{j+1}\\}$ connecting
$x^{\prime}_{j}$ to $x^{\prime}_{j+1}$.

Otherwise, for each $U=u(i)=u^{\prime}(j)$, apply 4.3.(3) (note that in this
case $U\subseteq O$) to find two disjoint arcs
$\gamma_{i},\gamma^{\prime}_{j}$ in
$U\cup\\{x_{i},x_{i+1},x^{\prime}_{j},x^{\prime}_{j+1}\\}$ connecting $x_{i}$
to $x_{i+1}$ and $x^{\prime}_{j}$ to $x^{\prime}_{j+1}$, respectively.

Let $\lambda,\lambda^{\prime}$ be the two disjoint arcs resulting from the
concatenations of the $\gamma_{i}$’s and $\gamma^{\prime}_{j}$’s,
respectively. Let $\tilde{\mathcal{V}}\preceq\mathcal{U}$ be a brick partition
fine enough that
$\operatorname{Star}(\lambda,\tilde{\mathcal{V}})\cap\operatorname{Star}(\lambda^{\prime},\tilde{\mathcal{V}})=\emptyset$.
Set
$P_{i}\coloneqq\operatorname{Star}(\lambda,\tilde{\mathcal{V}}(u(i))),P^{\prime}_{j}\coloneqq\operatorname{Star}(\lambda^{\prime},\tilde{\mathcal{V}}(u^{\prime}(j)))$,
for each $i<\operatorname{lh}(u),j<\operatorname{lh}(u^{\prime})$, and let
$\mathcal{V}$ be the amalgam of $\tilde{\mathcal{V}}$ refining $\mathcal{U}$
and containing $\bigvee P_{i}$ and $\bigvee P^{\prime}_{j}$, for all
$i<\operatorname{lh}(u),j<\operatorname{lh}(u^{\prime})$. Then
$v\coloneqq\langle P_{0},\dots,P_{\operatorname{lh}(u)-1}\rangle$ and
$v^{\prime}\coloneqq\langle
P^{\prime}_{0},\dots,P^{\prime}_{\operatorname{lh}(u^{\prime})-1}\rangle$ are
paths on $\mathcal{V}$ satisfying the conclusions of the proposition.

If $u$ is circular, we can take $x_{0}=x_{\operatorname{lh}(u)}$ in
$\operatorname{Cl}(w(-1))\cap\operatorname{Cl}(w(0))\cap(w(-1)\vee w(0))$, and
arcs $\gamma_{0},\gamma_{\operatorname{lh}(u)}$ connecting $x_{0}$ to
$x_{1},x_{\operatorname{lh}(u)-1}$, respectively, and add ‘$\\!\\!\text{
mod}\,\operatorname{lh}(u)$’ where needed. ∎

The  (Disjoint Path Property). allows us to find a robust cycle in any locally
non-planar open set.

######  Theorem 6.6.

Let $X$ be a Peano continuum with no locally separating points. If $X$ has a
locally non-planar open subset then it has a robust cycle.

###### Proof.

First let us show that it is sufficient to find
$(\mathcal{V}^{\prime},v^{\prime})\preceq(\mathcal{V},v)$ with $\mathcal{V}$ a
brick partition and $v^{\prime}$ a path, and a circular path $c$ disjoint form
$v^{\prime}$ satisfying (6.1) and (6.2) in Definition 6.3. Indeed, in this
case, by  (Path Doubling). (Proposition 6.1) we can assume that, up to a
refinement, there are two paths $v^{\prime}_{0},v^{\prime}_{1}$ which are
disjoint except in their endpoints, disjoint from $c$, and such that
$\rho^{\mathcal{V}^{\prime}}_{\mathcal{V}}\cdot v^{\prime}_{i}\sqsubseteq v$.
Let $P$ be a path from $v^{\prime}_{0}\cup v^{\prime}_{1}$ to $c$, and say
without loss of generality that it starts in $v^{\prime}_{0}$. Then the lasso
path that follows $v^{\prime}_{1}$ from beginning to end, then
$v^{\prime}_{0}$ backwards until meeting $P$, then follows $P$, then does a
complete lap following $c$, is as required.

We now proceed to find such $v^{\prime}$ and $c$. Suppose that $O\subseteq X$
is open and locally non-planar. Fix any brick partition $\mathcal{W}$ of $X$
which is fine enough that there are $W,W^{\prime}$ which are in the same
$E$-connected component of $\mathcal{W}(O)$ but are not adjacent. Let $w$ be a
path from $W$ to $W^{\prime}$ in $\mathcal{W}(O)$. By  (Path Doubling)., there
are a brick partition $\mathcal{S}\preceq\mathcal{W}$ and two paths
$s_{0},s_{1}$ on $\mathcal{S}$ which are disjoint except for their endpoints
and such that $\rho^{\mathcal{S}}_{\mathcal{W}}\cdot s_{i}$ monotonically
refines $w$, for $i<2$. Let $C$ be the circular path which follows $s_{0}$
forward and then $s_{1}$ backwards. Up to a refinement, we can assume $C$ is
spaced, by  (Spacing out Paths). (Lemma 5.3). Since $s_{0},s_{0}$ have length
at least $3$, $C$ has length at least $4$.

By  (Spacing out Paths). twice, we find a brick partition
$\mathcal{U}\preceq\mathcal{S}$ and a spaced circular path $D$ on
$\mathcal{U}$ such that $\rho^{\mathcal{U}}_{\mathcal{S}}\cdot D$
monotonically refines $C$ and

(6.11) $\rho^{\mathcal{U}}_{\mathcal{S}}([D]_{2})\subseteq C.$

Let $u$ be the empty walk and suppose that a brick partition and a walk
$(\mathcal{V},v)\preceq(\mathcal{U},u)$ are given. Up to a refinement, we can
suppose that $v$ is a path, by  (Upgrading Walks to Paths). (Theorem 6.2). Use
Lemma 3.1 to find a circular path $D^{\prime}$ on $\mathcal{V}$ such that
$\rho^{\mathcal{V}}_{\mathcal{U}}\cdot D^{\prime}$ monotonically refines $D$.
Since $D^{\prime}$ is confined in $\mathcal{V}(O)$, by  (Disjoint Path
Property). (Proposition 6.5) there are a brick partition
$\mathcal{V}^{\prime}\preceq\mathcal{V}$ and two disjoint paths $v^{\prime},c$
on $\mathcal{V}^{\prime}$ such that
$\rho^{\mathcal{V}^{\prime}}_{\mathcal{V}}\cdot
v_{0},\rho^{\mathcal{V}^{\prime}}_{\mathcal{V}}\cdot c$ monotonically refine
$v$ and $D^{\prime}$, respectively, and $c$ is circular. Then $c$ satisfies
(6.1) by (6.11), and it follows by Lemma 3.3 that it satisfies (6.2). ∎

###  6.3. Planar open set with non-locally separating simple closed curve

The next two results capture the essential aspects behind the assumption of
the existence of a non-locally separating simple closed curve in Theorem 1.2.
We start with an auxiliary definition.

######  Definition 6.7.

Let $X$ be a Peano continuum, $\gamma$ be a simple closed curve on $X$, and
$\mathcal{S}$ be a brick partition of $X$. A spaced circular path
$c=(c(0),\dots,c(\ell-1),c(0))$ on $\mathcal{S}$ is _$\gamma$ -compatible_ if

  1. (1)

$\\{c(i):i<\ell\\}=\operatorname{Star}(\gamma,\mathcal{S})$;

  2. (2)

$\\{c(i)\cap\gamma:i<\ell\\}$ is a brick partition of $\gamma$.

######  Lemma 6.8.

Let $X$ be a Peano continuum with no locally separating points and let
$\gamma$ be a non-locally separating simple closed curve in an open planar
subset $O\subseteq X$. For each brick partition $\mathcal{U}$ of $X$, there
are a brick partition $\mathcal{V}\preceq\mathcal{U}$ and a
$\gamma$-compatible spaced circular path $c$ on $\mathcal{V}$.

###### Proof.

Up to a refinement we can assume that if
$U\in\operatorname{Star}(\gamma,\mathcal{U})$, then
$\operatorname{Cl}(U)\subseteq O$. By Jordan’s theorem, we can embed $O$ in
the unit disk in the plane in such a way that $\gamma$ is mapped to the unit
circle, since $\gamma$ is non-separating in $O$. Fix the usual
$\mathbb{R}^{2}$ metric.

######  Claim 6.8.1.

For each $U\in\operatorname{Star}(\gamma,\mathcal{U})$, it holds that
$\gamma\cap\operatorname{Cl}(U)$ has finitely many connected components, all
of which are non-trivial. Moreover $\gamma\cap\operatorname{Bd}(U)$ is totally
disconnected.

###### Proof of the Claim.

Let $\varepsilon$ be less than half of
$\min\\{\operatorname{diam}(U):U\in\mathcal{U}\\}$ and let $\delta$ be the
uniform local connectedness constant of $U$ relative to $\varepsilon$. We
prove that if two points $y_{0},y_{1}\in\gamma\cap\operatorname{Cl}(U)$ have
distance less than $\delta$, then they belong to the same connected component.
Indeed, in this case there is an arc $\lambda$ from $y_{0}$ to $y_{1}$ in
$U\cup\\{y_{0},y_{1}\\}$ of diameter less than $\varepsilon$. The arc
$\lambda$ separates $O$ in two regions, one of which meets $U$, has diameter
less that $2\varepsilon$, and contains in its interior an open interval
$I\subseteq\gamma$ with endpoints $y_{0},y_{1}$. But then such region is too
small to contain any $U^{\prime}\in\mathcal{U}$. Moreover, it cannot even meet
any $U^{\prime}\neq U$, since $\lambda$ would separate $U^{\prime}$ and
$\lambda$ is contained in $\operatorname{Cl}(U)$. Therefore it has to be
contained in $\operatorname{Cl}(U)$. But then $I\subseteq U$, and thus
$y_{0},y_{1}$ are in the same connected component of
$\gamma\cap\operatorname{Cl}(U)$. This shows that there are only finitely many
components. It also shows that $\gamma\cap\operatorname{Bd}(U)$ is totally
disconnected, since the open interval between any two points in the same
connected component of $\gamma\cap\operatorname{Cl}(U)$ is contained in $U$.

Suppose towards a contradiction that $\\{x\\}$ is a trivial connected
component of $\operatorname{Cl}(U)\cap\gamma$, so in particular
$x\in\operatorname{Bd}(U)$. Let $\delta_{0}$ be the minimum between $\delta$
and the uniform local connectedness constant of
$\bigvee\operatorname{Star}(x,\mathcal{U}\setminus\\{U\\})$ relative to
$\varepsilon$. Take two points
$y_{0},y_{1}\in\gamma\cap\operatorname{Star}(x,\mathcal{U}\setminus\\{U\\})$
whose distance is smaller than $\delta_{0}$ and such that
$x\in\gamma(y_{0},y_{1})$. Then there is an arc $\lambda$ of diameter less
than $\varepsilon$ connecting $y_{0},y_{1}$ in
$\bigvee\operatorname{Star}(x,\mathcal{U}\setminus\\{U\\})$. In particular
$\lambda\subseteq O$, so it separates $O$ in two regions, one of which
contains $x$ in its interior and has diameter less that $2\varepsilon$.
Arguing as in the first part of the proof we conclude that $U$ cannot be
contained in such region, so $x\not\in\operatorname{Cl}(U)$, a contradiction.
∎

Let $\gamma_{0},\dots,\gamma_{n-1}$ be the collection of connected components
of $\gamma$ in $\operatorname{Cl}(U)$, for $U$ ranging in $\mathcal{U}$. Any
three of them have empty intersection, since otherwise $\gamma$ would contain
a triod. Moreover, if $i\neq j$, then
$\lvert\gamma_{i}\cap\gamma_{j}\rvert\leq 1$. Indeed, since they are arcs in
the same circle, and $n\geq 3$, their intersection can only by empty, a point,
or a nontrivial arc. But such an intersection is contained in some
$\operatorname{Bd}(U)$, and thus is totally disconnected. It follows that
their interior in $\gamma$ form a brick partition of $\gamma$, and we can
assume that they are ordered so that
$\gamma_{0}\mathbin{E}\gamma_{1}\mathbin{E}\cdots\mathbin{E}\gamma_{n-1}\mathbin{E}\gamma_{0}$.

Let $\mathcal{V}^{\prime}\preceq\mathcal{U}$ be a brick partition fine enough
that $\operatorname{Star}(\gamma_{i},\mathcal{V}^{\prime})$ and
$\operatorname{Star}(\gamma_{j},\mathcal{V}^{\prime})$ have distance at least
2 in $\mathcal{V}^{\prime}$, for any $\lvert i-j\rvert>1\text{ mod}\,n$. Let
$\mathcal{V}$ be an amalgam of $\mathcal{V}^{\prime}$ refining $\mathcal{S}$
and containing
$P_{i}\coloneqq\bigvee\operatorname{Star}(\gamma_{i},\mathcal{V}^{\prime}(U))$,
where $U\in\mathcal{U}$ is such that $\gamma_{i}\subseteq\operatorname{Cl}(U)$
and $i<n$. Then $c\coloneqq\langle P_{0},\dots,P_{n-1},P_{0}\rangle$ is a
$\gamma$-compatible spaced circular walk. ∎

It is now not hard to see that we can zoom-in and disjoin any path from a
$\gamma$-compatible spaced circular path.

######  Proposition 6.9.

Let $X$ be a Peano continuum with no locally separating points and let
$\gamma$ be a non-locally separating simple closed curve contained in an open
planar subset $O\subseteq X$. For each brick partition $\mathcal{V}$ of $X$
and path $v$ on $\mathcal{V}$, there are a brick partition and a path
$(\mathcal{V}^{\prime},v^{\prime})\preceq(\mathcal{V},v)$, and a
$\gamma$-compatible spaced circular path $c$ on $\mathcal{V}^{\prime}$
disjoint from $v^{\prime}$.

###### Proof.

For each $1<i<\operatorname{lh}(v)-1$, let $x_{i}$ be a point in
$\operatorname{Cl}(v(i-1))\cap\operatorname{Cl}(v(i))\cap(v(i-1)\vee v(i))$
which does not belong to $\gamma$, which exist by 4.2.(3). Let $\lambda_{i}$
be an arc in $v(i)\cup\\{x_{i-1},x_{i}\\}$ connecting $x_{i-1}$ to $x_{i}$ and
disjoint from $\gamma$, which exist by 4.3.(4).

Let $\lambda$ be the concatenation of the $\lambda_{i}$’s and let
$\widetilde{\mathcal{V}}\preceq\mathcal{V}$ be a brick partition fine enough
that

(6.12)
$\operatorname{Star}(\lambda,\widetilde{\mathcal{V}})\cap\operatorname{Star}(\gamma,\widetilde{\mathcal{V}})=\emptyset,$

and such that there is a $\gamma$-compatible spaced circular path $c$ on
$\widetilde{\mathcal{V}}$, which is possible by Lemma 6.8. Notice that $c$ is
confined to $\operatorname{Star}(\gamma,\widetilde{\mathcal{V}})$.

Set
$P_{j}\coloneqq\bigvee\operatorname{Star}(\lambda,\widetilde{\mathcal{V}}(v(j)))$
for each $j<\operatorname{lh}(v)$, and let $\mathcal{V}^{\prime}$ be any
amalgam of $\widetilde{\mathcal{V}}$ refining $\mathcal{V}$ which contains all
$P_{j}$ and all elements of $c$. Then $v^{\prime}\coloneqq\langle
P_{0},\dots,P_{\operatorname{lh}(v)-1}\rangle$ is a path that is disjoint from
$c$ by (6.12). ∎

We are ready to prove that $\gamma$-compatible spaced circular paths are
robust.

######  Theorem 6.10.

Let $X$ be a Peano continuum with no locally separating points. If $X$ has a
planar open set containing a simple closed curve which is not locally
separating then it has a robust cycle.

###### Proof.

As in Theorem 6.6, it is sufficient to find a brick partition and a path
$(\mathcal{V}^{\prime},v^{\prime})\preceq(\mathcal{V},v)$ and a circular path
$c$ disjoint form $v^{\prime}$ satisfying the conditions in Definition 6.3.

Let $O\subseteq X$ be a planar open set and $\gamma\subseteq O$ a simple
closed curve which is not locally separating. Fix any brick partition
$\mathcal{W}$ of $X$ which is fine enough that
$\lvert\operatorname{Star}(\gamma,\mathcal{W})\rvert\geq 4$. Let
$\mathcal{S}\preceq\mathcal{W}$ be a brick partition and let $C$ be
$\gamma$-compatible spaced circular path $C$ on $\mathcal{S}$, using Lemma
6.8. By the assumption on $\mathcal{W}$, the length $\ell$ of $C$ is at least
$4$.

Let $\mathcal{U}_{0}\preceq\mathcal{S}$ be a brick partition fine enough that

(6.13)
$d_{\mathcal{U}_{0}}(\operatorname{Star}(\gamma,\mathcal{U}_{0}),\mathcal{U}_{0}(\mathcal{S}\setminus
C))>2.$

By Lemma 6.8, there are a brick partition $\mathcal{U}\preceq\mathcal{U}_{0}$
and a $\gamma$-compatible spaced circular path $D$ on $\mathcal{U}$. Let $u$
be the empty walk on $\mathcal{U}$.

Suppose a brick partition and a walk $(\mathcal{V},v)\preceq(\mathcal{U},u)$
are given. Up to a refinement, we can suppose that $v$ is a path, by
(Upgrading Walks to Paths). (Theorem 6.2). By Proposition 6.9, there are a
brick partition and a path
$(\mathcal{V}^{\prime},v^{\prime})\preceq(\mathcal{V},v)$ and a
$\gamma$-compatible spaced circular path $c$ on $\mathcal{V}^{\prime}$ which
is disjoint from $v^{\prime}$.

Now $\rho^{\mathcal{V}^{\prime}}_{\mathcal{U}}\cdot c$ is confined in $D$, by
point 1 of Definition 6.7, so (6.1) follows from (6.13). Since changing the
starting vertex of $C$ does not affect the winding number, and flipping the
direction of $C$ just flips the sign of
$\operatorname{wd}_{C}(\rho^{\mathcal{V}^{\prime}}_{\mathcal{S}}\cdot c)$, we
can assume that $c(0)\in\mathcal{V}^{\prime}(C(0))$ and that the first time
$c$ exits $\mathcal{V}^{\prime}(C(0))$ it enters $\mathcal{V}^{\prime}(C(1))$.
Item (2) in Definition 6.7 then implies that both $c$ and $C$ induce brick
partitions of $\gamma$, with the former refining the latter. It follows that
$\rho^{\mathcal{V}^{\prime}}_{\mathcal{S}}\cdot c$ monotonically refines $C$,
thus by Lemma 3.3 we can conclude that
$\lvert\operatorname{wd}_{C}(\rho^{\mathcal{V}^{\prime}}_{\mathcal{S}}\cdot
c)\rvert=\ell$. ∎

###  6.4. Circular covering

We recall from the introduction that a Peano continuum $X$ has a circular
covering if it admits a covering $O_{0},\dots,O_{\ell-1}$ in $\ell\geq 4$
connected open subsets such that
$\operatorname{Cl}(O_{i})\cap\operatorname{Cl}(O_{j})\neq\emptyset$ if and
only if $\lvert i-j\rvert\leq 1\text{ mod}\,\ell$, and
$O_{i}\setminus\bigcup_{j\neq i}O_{j}$ is connected for each $i<\ell$.

######  Proposition 6.11.

A Peano continuum has a circular covering if and only if it admits a brick
partition whose nerve graph is a cycle of length at least $4$.

###### Proof.

Suppose that $X$ has a brick partition $\mathcal{S}$ which is a cycle of
length $\ell\geq 4$. Say the elements of $\mathcal{S}$ are
$S_{0}\mathbin{E}S_{1}\mathbin{E}\cdots\mathbin{E}S_{\ell-1}\mathbin{E}S_{0}$.
Let $\mathcal{W}\preceq\mathcal{S}$ be given by Proposition 5.5 applied to
$\mathcal{U}=\mathcal{V}=\mathcal{S}$. We can moreover pick $\mathcal{W}$ fine
enough so that no element of
$\operatorname{Star}(\operatorname{Cl}(S_{i}),\mathcal{W})$ is $E$-adjacent to
one of $\operatorname{Star}(\operatorname{Cl}(S_{i+2\text{
mod}\,\ell}),\mathcal{U})$ for every $i<\ell$. Note furthermore that every
$\operatorname{Star}(\operatorname{Cl}(S_{i}),\mathcal{W})$ is $E$-connected,
so setting

$O_{i}\coloneqq\bigvee\operatorname{Star}(\operatorname{Cl}(S_{i}),\mathcal{W}),$

we get a covering $O_{0},\dots,O_{\ell-1}$ whose pieces are connected and
intersect as required. Moreover, $O_{i}\setminus\bigcup_{j\neq
i}O_{j}=\bigvee\operatorname{Core}(S_{i},\mathcal{W})$, which is connected by
Proposition 5.5.

Conversely, suppose $X$ has a circular covering $O_{0},\dots,O_{\ell-1}$.
Define

$C_{i}\coloneqq\operatorname{Cl}(O_{i})\setminus\bigcup_{j\neq i}O_{j}.$

These sets are pairwise disjoint, since $x\in C_{j}\cap C_{k}$ with $j\not=k$
if and only if $x\not\in O_{i}$ for every $i<\ell$, which is not possible.

Let $\mathcal{U}$ be a brick partition refining
$\\{O_{0},\dots,O_{\ell-1}\\}$, which is fine enough that

(6.14)
$\operatorname{Star}(C_{i},\mathcal{U})\cap\operatorname{Star}(C_{j},\mathcal{U})=\emptyset\text{
for each }i\neq j,$

and

(6.15) $\operatorname{Star}(O_{i}\cap O_{i+1\text{
mod}\,\ell},\mathcal{U})\cap\operatorname{Star}(O_{j}\cap O_{j+1\text{
mod}\,\ell},\mathcal{U})=\emptyset\text{ for each }i\neq j.$

Since $C_{i}$ is connected,
$\mathcal{U}_{i}=\operatorname{Star}(C_{i},\mathcal{U})$ is $E$-connected.
Given $i<\ell$, divide $\operatorname{Star}(O_{i}\cap O_{i+1\text{
mod}\,\ell},\mathcal{U})$ into two sets
$\mathcal{P}_{i},\mathcal{Q}_{i+1\text{ mod}\,\ell}$ based on whether $U$ is
such that

$d_{\mathcal{U}}(U,\mathcal{U}_{i})\leq
d_{\mathcal{U}}(U,\mathcal{U}_{i+1\text{ mod}\,\ell})\text{ or
}d_{\mathcal{U}}(U,\mathcal{U}_{i+1\text{
mod}\,\ell})<d_{\mathcal{U}}(U,\mathcal{U}_{i}).$

Using (6.14) and (6.15) we see that if $U\in\mathcal{P}_{i}$ then all the
elements of any of the shortest paths connecting it to $\mathcal{U}_{i}$ are
either in $\mathcal{U}_{i}$, or in $\operatorname{Star}(O_{i}\cap O_{i+1\text{
mod}\,\ell},\mathcal{U})$, and therefore in $\mathcal{P}_{i}$. A similar
observation holds for $\mathcal{Q}_{i}$.

We can thus conclude that the sets
$\mathcal{Q}_{i}\cup\mathcal{U}_{i}\cup\mathcal{P}_{i}$ are $E$-connected and
form a partition of $\mathcal{U}$. The amalgam $\mathcal{S}$ of $\mathcal{U}$
obtained from such partition is the desired cycle of $\ell$ elements. ∎

We now show that the circular brick partition found in Proposition 6.11 is
robust.

######  Theorem 6.12.

Let $X$ be a Peano continuum with no locally separating points. If $X$ has a
circular covering then it has a robust cycle.

###### Proof.

By Proposition 6.11, there is a brick partition $\mathcal{S}$ of $X$ which is
a cycle of length $\ell\geq 4$. Let $C$ be a circular path around
$\mathcal{S}$ (in particular, $\operatorname{lh}(C)=\ell$). By  (Upgrading
Walks to Paths). (Theorem 6.2), there are a brick partition
$\mathcal{U}\preceq\mathcal{S}$ and a spaced path $u$ such that
$\rho^{\mathcal{U}}_{\mathcal{S}}\cdot u$ monotonically refines
$C{}^{\frown}C$, which is possible since $C{}^{\frown}C$ is uncrossed. In
particular, $\operatorname{wd}_{C}(\rho^{\mathcal{U}}_{\mathcal{S}}\cdot
u)=2\ell$, by Lemma 3.3.

Suppose a brick partition and a walk $(\mathcal{V},v)\preceq(\mathcal{U},u)$
are given. Suppose without loss on generality that $v(0)\in\mathcal{V}(C(0))$.
Note that for any initial segment $t$ of $v$ such that
$\rho^{\mathcal{V}}_{\mathcal{S}}(t(-1))\in C(k)$ we get

(6.16) $\operatorname{wd}_{C}(\rho^{\mathcal{V}}_{\mathcal{S}}\cdot t)\equiv
k\text{ mod}\,\ell.$

Let $t_{0}$ be the shortest initial segment of $v$ such that
$\lvert\operatorname{wd}_{C}(\rho^{\mathcal{V}}_{\mathcal{S}}\cdot
t_{0})\rvert>\ell$. Such initial segment exists by Lemma 3.4 applied to $v$
and $u$, and let $k_{0}<\ell$ be such that
$\rho^{\mathcal{V}}_{\mathcal{S}}(t_{0}(-1))\in C(k_{0})$; then (6.16) implies
$k_{0}>0$.

Using  (Path Doubling). (Proposition 6.1) we find a brick partition
$\mathcal{V}^{\prime}\preceq\mathcal{V}$ and two paths $v_{0},v_{1}$ which are
disjoint except at their ends and such that
$\rho^{\mathcal{V}^{\prime}}_{\mathcal{V}}\cdot v_{i}$ monotonically refines
$v$, for $i<2$. Then Lemma 3.3 gives

(6.17) $\operatorname{wd}_{C}(\rho^{\mathcal{V}^{\prime}}_{\mathcal{S}}\cdot
v_{i})=\operatorname{wd}_{C}(\rho^{\mathcal{V}}_{\mathcal{S}}\cdot v),\text{
for }i=0,1.$

Let $s_{i}$ be an initial segment of $v_{i}$ such that
$\rho^{\mathcal{V}^{\prime}}_{\mathcal{V}}\cdot s_{i}$ monotonically refines
$t$. Then, again by Lemma 3.3,

$\operatorname{wd}_{C}(\rho^{\mathcal{V}^{\prime}}_{\mathcal{S}}\cdot
s_{i})=\operatorname{wd}_{C}(\rho^{\mathcal{V}}_{\mathcal{S}}\cdot t)>\ell.$

To lighten notation, let us refer to
$\operatorname{wd}_{C}(\rho^{\mathcal{V}^{\prime}}_{\mathcal{U}}\cdot r)$ as
the winding number of $r$, for any walk $r$ on $\mathcal{V}^{\prime}$.

Let us partition the set of the $V$’s in $\mathcal{V}^{\prime}(C(k_{0}))$
which belong to $v_{0}\cup v_{1}$ into two sets $A,B$, depending on whether
the winding number of the corresponding initial segment up to $V$ is smaller
or greater than $\ell$, in absolute value (by (6.16) the value $\ell$ is never
achieved). Both of these sets are nonempty, as witnessed by the $s_{i}$’s. Let
$z$ be a path from $A$ to $B$ in $\mathcal{V}^{\prime}(C(k_{0}))$. There are
two cases.

Case 1: There is $i<2$ such that both endpoints of $z$ are contained in
$v_{i}$, and we might as well assume $i=1$. Let $v^{\prime}{}^{\frown}c$ be
the lasso path which follows $v_{0}$ from beginning to end, then $v_{1}$
backwards until it hits $z$ for the first time (this is $v^{\prime}$), then
follows $z$ (possibly backwards), then $v_{1}$ forwards until it hits $z$
again (this is $c$). Since $z$ is included in $\mathcal{V}^{\prime}(C(k_{0}))$
it does not contribute to the winding number of $c$, which is therefore equal
to the difference between the winding numbers of the initial segments of
$v_{1}$ at either end of $z$, which are equal modulo $\ell$ by (6.16). But
these are in $A$ and $B$, respectively, so the absolute value of the
difference is greater or equal than $\ell$, and (6.2), is satisfied.

Case 2: Otherwise, let $v^{\prime}{}^{\frown}c$ be the lasso path which
follows $v_{0}$ from the beginning until it hits $z$ (this is $v^{\prime}$),
then continues following $v_{0}$ until the end, then $v_{1}$ backwards until
it hits $z$, then follows $z$, possibly backwards (this is $c$). As above, $z$
does not contribute to the winding number of $c$, which is therefore equal to
the difference between the winding numbers of the final segments of $v_{1}$
and $v_{0}$ at either end of $z$. But since $v_{0},v_{1}$ have the same
winding number by (6.17), the difference between the winding numbers of the
final segments is equal to the difference of the winding numbers of the
respective initial segments. We therefore conclude as in Case 1.

In both cases, $v_{0}$ is an initial segment of $v^{\prime}{}^{\frown}c$, so
$(\mathcal{V}^{\prime},v^{\prime}{}^{\frown}c)\preceq(\mathcal{V},v)$.
Moreover, (6.1) is trivially true, since $C$ covers all of $\mathcal{S}$. ∎

We close this section showing that the sphere and the real projective plane
are the only two closed surfaces without a circular covering.

A continuum is _unicoherent_ if it cannot be written as the union of two
subcontinua whose intersection is disconnected. If $X$ has a circular covering
covering $O_{0},\dots,O_{\ell-1}$, it is not unicoherent, as witnessed by
$\operatorname{Cl}(O_{0})$ and $\bigcup_{i\neq 0}\operatorname{Cl}(O_{i})$.

######  Proposition 6.13.

The only two closed surfaces without a circular covering are the sphere and
the real projective plane.

###### Proof.

By [MR0007095]*Theorem 7.4, the sphere and the real projective plane are
unicoherent, therefore they do not have a circular covering.

On the other hand, it is clear that the torus and the Klein bottle have
circular coverings. We conclude by showing that the connected sum of a surface
$X$ which has a circular covering, with any other surface $X^{\prime}$, has a
circular covering: suppose $O_{0},\dots,O_{\ell-1}$ is a circular covering of
$X$; take an open disk $D$ in $O_{0}\setminus\bigcup_{j\neq 0}O_{j}$, and any
open disk $D^{\prime}$ in $X^{\prime}$, and glue $X,X^{\prime}$ along
$\operatorname{Bd}(D),\operatorname{Bd}(D^{\prime})$. Then $(O_{0}\setminus
D)\cup(X^{\prime}\setminus D^{\prime}),O_{1},\dots,O_{\ell-1}$ is a circular
covering of $X\\#X^{\prime}$. ∎

##  7\. Classifiability and Dynamics

In this section we elaborate on the dynamical consequences of Theorem 1.2 and
we derive from it, under sufficient homogeneity conditions, Theorem 1.3,
Corollary 1.7 and Corollary 1.8 (see Corollaries 7.8 and 7.9).

Recall that a Polish group has the generic point property if every minimal
flow has a comeager orbit. Given a Peano continuum $X$ and a subgroup
$G\leq\operatorname{Homeo}(X)$, we can obtain the failure of the generic point
property for $G$ directly from Theorem 1.2, if we can prove that the action
$G\curvearrowright\Phi(X)$ is minimal. The first thing to notice is that a
necessary prerequisite for this is the minimality the action
$G\curvearrowright X$, since we recall that the map from $\Phi(X)$ onto $X$
associating to each chain $\mathcal{C}$ its root $\bigcap\mathcal{C}$ is a
continuous $\operatorname{Homeo}(X)$-equivariant map. Therefore, for any
compact manifold with boundary, the action on the space of chains is not
minimal. Minimality of $G\curvearrowright X$ is not sufficient, though: while
$\operatorname{Homeo}(S^{1})$ acts minimally, even transitively on the circle
$S^{1}$, $\operatorname{Homeo}(S^{1})\curvearrowright\Phi(X)$ has a proper
closed invariant subset: take the set of those chains $\mathcal{C}$ such that
for each $C\in\mathcal{C}\setminus\\{S^{1}\\}$ the root of $\mathcal{C}$
belongs to the boundary of $C$.

The following gives a sufficient condition for minimality of
$G\curvearrowright\Phi(X)$, when $X$ is a Peano continuum with no locally
separating points.

######  Proposition 7.1.

Let $X$ be a Peano continuum with no locally separating points. Let
$G\leq\operatorname{Homeo}(X)$ be such that $G\curvearrowright X$ is minimal.
Suppose that for each $\mathcal{C}\in\Phi(X)$ with root $x_{0}$, and any
connected open sets $U_{0},U_{1},V$ with $x_{0}\in U_{0}$ and
$U_{0},U_{1}\subseteq V$, there are $C\in\mathcal{C}$ and $g\in G$ such that:

  1. (1)

$g(x_{0})\in U_{0}$,

  2. (2)

$g[C]\cap U_{1}\neq\emptyset$,

  3. (3)

$g[C]\subseteq V$.

Then the action $G\curvearrowright\Phi(X)$ is minimal.

The proof relies on the following lemma, whose straightforward proof is left
to the reader.

######  Lemma 7.2.

Let $X$ be a Peano continuum, let $\mathcal{U}$ be a brick partition of $X$
and let $u$ be a spaced path on $\mathcal{U}$. Let $\mathcal{C}\in\Phi(X)$
with root $x_{0}$ be such that $x_{0}\in u(0)$ and suppose there is
$C\in\mathcal{C}$ such that $C\cap u(-1)\neq\emptyset$ and
$C\subseteq\bigvee_{i<\operatorname{lh}(u)}u(i)$. Then
$\mathcal{C}\in\mathbf{O}_{\mathcal{U},u}$.

###### Proof of Proposition 7.1.

Let $\mathcal{C}\in\Phi(X)$ with root $x_{0}$ and a nonempty open
$\mathbf{U}\subseteq\Phi(X)$ be given. By Proposition 5.2 and  (Upgrading
Walks to Paths). (Theorem 6.2), there are a brick partition $\mathcal{U}$ of
$X$ and a spaced path $u$ on $\mathcal{U}$ such that
$\mathbf{O}_{\mathcal{U},u}\subseteq\mathbf{U}$.

Since $G$ acts minimally on $X$, there is $g_{0}\in G$ such that
$g_{0}(x_{0})\in u(0)$. By hypothesis there are $g_{0}[C]\in
g_{0}\cdot\mathcal{C}$ and $g_{1}\in G$ such that $g_{1}g_{0}(x_{0})\in u(0)$,
$g_{1}g_{0}[C]\cap u(-1)\neq\emptyset$ and
$g_{1}g_{0}[C]\subseteq\bigvee_{i<\operatorname{lh}(u)}u(i)$. We conclude that
$g_{1}g_{0}\cdot\mathcal{C}\in\mathbf{O}_{\mathcal{U},u}$, by Lemma 7.2. ∎

We briefly recall what it means for an equivalence relation on a Polish space
to be classifiable by countable structures and refer the reader to
[hjorth2000classification]*Section 2.3 for more details. Given a countable
relational language $\mathcal{L}=\\{R_{i}\\}_{i\in I}$, where each $R_{i}$ has
arity $n_{i}$, we topologize the collection $\mathrm{Mod}(\mathcal{L})$ of all
$\mathcal{L}$-structures with domain $\mathbb{N}$, by identifying it with the
product space

$\prod_{i\in I}2^{\mathbb{N}^{n(i)}}$

where every $R_{i}\subseteq\mathbb{N}^{n(i)}$ is identified with its indicator
function. Consider now another Polish space $Y$ and an equivalence relation
$\equiv$ on $Y$. We say that $(Y,\equiv)$ is _classifiable by countable
structures_ if and only if there exists a Borel function $f\colon
Y\to\mathrm{Mod}(\mathcal{L})$, for some countable relational language
$\mathcal{L}$, such that for all $y,y^{\prime}\in Y$,

(7.1) $y\equiv y^{\prime}\iff f(y)\cong f(y^{\prime}).$

In order to prove Corollary 1.8, we rely on the notion of turbulence and on a
classic theorem due to Hjorth.

######  Definition 7.3 ([hjorth2000classification]).

Let $G\curvearrowright Y$ be a continuous action on a topological space. Given
a point $y\in Y$, an open neighborhood $\mathbf{U}$ of $y$, and an open
neighborhood $G_{0}$ of the identity in $G$, the _$(\mathbf{U},G_{0})$ -local
orbit of $y$_ is the set of all $y^{\prime}\in\mathbf{U}$ such that there is a
finite sequence $y=y_{0},\dots,y_{n}=y^{\prime}$ of elements of $\mathbf{U}$
verifying $y_{i+1}\in G_{0}\cdot y_{i}$ for all $i<n$. We shall say that _all
local orbits are somewhere dense_ if for any $y,\mathbf{U},G_{0}$ as above,
the closure of the $(\mathbf{U},G_{0})$-local orbit of $y$ has nonempty
interior.

An action is _turbulent_ if it is minimal, it has no comeager orbit and all
local orbits are somewhere dense.

######  Theorem 7.4 ([hjorth2000classification]*Corollary 3.19).

Let $G$ be a Polish group and $G\curvearrowright Y$ be a continuous action on
a Polish space. If the action is turbulent then the equivalence relation
$y\equiv_{G}y^{\prime}$ if and only if $y^{\prime}\in G\cdot y$ is not
classifiable by countable structures.

###  7.1. Strongly Locally Homogeneous Spaces

In Theorem 7.7 below we show that the action $G\curvearrowright\Phi(X)$ is
minimal and all local orbits are somewhere dense, as long as $X$ satisfies a
strong form of homogeneity and $G\leq\operatorname{Homeo}(X)$ is large enough.
We begin by making these last two conditions precise.

For $G\leq\operatorname{Homeo}(X)$, a continuous action $G\curvearrowright X$
is called _locally transitive_ 111Notice that in [gutman_minimal] local
transitivity refers to a slightly stronger property. Nevertheless, the two
properties are equivalent when $X$ is a Peano continuum with no locally
separating points. if for any $x\in X$ and any neighborhood $V$ of $x$,
$G_{V}\cdot x$ contains an open neighborhood of $x$, where

(7.2) $G_{V}\coloneqq\\{g\in G:g(y)=y\text{ for all }y\not\in V\\}$

is the _rigid stabilizer_ of $V$ in $G$.

Spaces $X$ such that $\operatorname{Homeo}(X)\curvearrowright X$ is locally
transitive are called _strongly locally homogeneous_ (see for instance
[MR2674033]). All closed manifolds, as well as the Hilbert cube, are strongly
locally homogeneous. Furthermore, $\operatorname{Homeo}_{0}(X)$ acts locally
transitively on $X$ whenever $X$ is a closed manifold, as do
$\operatorname{Homeo}_{+}(X)$ and $\mathrm{Diffeo}(X)$ when $X$ is moreover
orientable or smooth, respectively [gutman_minimal]*Examples 3.1. Other
examples of strongly locally homogeneous Peano continua are the Menger curve
$\mu^{1}$ and the universal $k$-dimensional compactum $\mu^{k}$, for $k>1$
[MR0920964]*Theorem 3.2.2.

######  Proposition 7.5 ([MR2674033]*Proposition 7).

If $X$ is a Peano continuum with no locally separating points and
$G\leq\operatorname{Homeo}(X)$ is such that $G\curvearrowright X$ is locally
transitive, then for any connected open set $V\subseteq X$ and any bijection
$f\colon A\to B$ between finite subsets of $V$, there is $g\in G_{V}$
extending $f$. In particular, $G$ acts on $X$ transitively.

As a consequence, strongly locally homogeneous Peano continuum with no locally
separating points are homogeneous. Any Peano continuum which is homogeneous
and is not the circle or a closed surface, is locally non-planar
[MR2674033]*Remark 8. It then follows from Theorem 6.4 and Theorem 6.6 that:

######  Corollary 7.6.

No homogeneous Peano continuum except for the circle and, possibly, the sphere
and real projective plane, has a generic chain.

We now turn to the main theorem of this section. We remark that minimality of
$G\curvearrowright\Phi(X)$, for $X$ a closed manifold of dimension at least 2
and $G\leq\operatorname{Homeo}(X)$ a subgroup which acts locally transitively
on $X$, was already proved in [gutman_minimal]*Theorem 6.5.

######  Theorem 7.7.

Let $X$ be a Peano continuum with no locally separating points. Let
$G\leq\operatorname{Homeo}(X)$ be such that $G\curvearrowright X$ is locally
transitive. Then the action $G\curvearrowright\Phi(X)$ is minimal and all
local orbits are somewhere dense.

###### Proof.

By Proposition 7.5, $G\curvearrowright X$ is minimal, so to prove minimality
of $G\curvearrowright\Phi(X)$ it is enough to check that the conditions of
Proposition 7.1 are satisfied. To this end, let $\mathcal{C}\in\Phi(X)$ with
root $x_{0}$ be given, together with open connected sets $U_{0},U_{1},V$ such
that $x_{0}\in U_{0}$ and $U_{0},U_{1}\subseteq V$.

Let $C\in\mathcal{C}$ be non-trivial and such that $C\subseteq V$, and pick
$r_{0}\in C\setminus\\{x_{0}\\}$. Fix moreover $r_{1}\in U_{1}$. By
Proposition 7.5, there is $g\in G_{V}$ such that $g(x_{0})=x_{0}\in U_{0}$ and
$g(r_{0})=r_{1}\in U_{1}\cap g[C]$. As $g$ is the identity outside $V$, we get
$g[C]\subseteq V$, hence the action $G\curvearrowright\Phi(X)$ is minimal by
Proposition 7.1.

Towards proving that all local orbits are somewhere dense, fix
$\mathcal{C}\in\Phi(X)$ with root $x_{0}$, an open neighborhood $\mathbf{U}$
of $\mathcal{C}$ and an open neighborhood $G_{0}$ of the identity in $G$. By
Proposition 5.2 and  (Upgrading Walks to Paths). (Theorem 6.2), there are a
brick partition $\mathcal{U}$ and a spaced path $u$ on $\mathcal{U}$ such that
$\mathbf{O}_{\mathcal{U},u}\subseteq\mathbf{U}$.

We prove that the closure of the $(\mathbf{U},G_{0})$-local orbit of
$\mathcal{C}$ contains $\mathbf{O}_{\mathcal{U},u}$. To this end, let
$\mathbf{V}\subseteq\mathbf{O}_{\mathcal{U},u}$ be open and nonempty. By
Proposition 5.2, there are a brick partition and a walk
$(\mathcal{V},v)\preceq(\mathcal{U},u)$ such that
$\mathbf{O}_{\mathcal{V},v}\subseteq\mathbf{V}$. Up to a refinement, by
(Upgrading Walks to Paths)., we can suppose that $v$ is a spaced path, and
that $\mathcal{V}$ is fine enough that

$G_{\mathcal{V},1}\subseteq G_{0},$

where $G_{\mathcal{V},1}$ is defined in (5.6). Crucially, for any
$V\mathbin{E}V^{\prime}$ in $\mathcal{V}$, the rigid stabilizer $G_{V\vee
V^{\prime}}$ (see (7.2)) is a subset of $G_{\mathcal{V},1}$, and hence of
$G_{0}$. We use this fact repeatedly from here onwards.

By picking $\mathcal{V}$ fine enough we can moreover assume there is
$V_{0}\in\mathcal{V}(u(0))$ such that $x_{0}\in\operatorname{Cl}(V_{0})$ and
such that $[V_{0}]_{1}\subseteq u(0)$. Fix a path $P$ from $V_{0}$ to $v(0)$
in $\mathcal{V}(u(0))$. We will move $x_{0}$ along $P$, step by step. As a
preliminary move, let $\mathcal{C}_{-1}\coloneqq\mathcal{C}$ and, using
Proposition 7.5, find $g_{-1}\in G_{[V_{0}]_{1}}\subseteq G_{\mathcal{V},1}$
mapping $x_{0}$ to some $y_{0}\in V_{0}=P(0)$ and set
$\mathcal{C}_{0}\coloneqq g_{-1}\cdot\mathcal{C}$. Note that
$\mathcal{C}_{0}\in\mathbf{O}_{\mathcal{U},u}$ since $g_{-1}$ fixes every
element of $u$ setwise. Suppose next we are given
$\mathcal{C}_{i}\in\mathbf{O}_{\mathcal{U},u}$ with root $y_{i}\in P(i)$, for
some $i<\operatorname{lh}(P)-1$. By Proposition 7.5, there is $g_{i}\in
G_{P(i)\vee P(i+1)}$ which maps $y_{i}$ to some point $y_{i+1}\in P(i+1)$.
Again, since $g_{i}$ fixes every element of $u$ setwise,
$\mathcal{C}_{i+1}\coloneqq g_{i}\cdot\mathcal{C}_{i}$ belongs to
$\mathbf{O}_{\mathcal{U},u}$.

To ease the notation, let
$\mathcal{D}_{0}\coloneqq\mathcal{C}_{\operatorname{lh}(P)-1}$ and
$z_{0}\coloneqq x_{\operatorname{lh}(P)-1}$. Then $z_{0}$ is the root of
$\mathcal{D}_{0}$ and it belongs to $v(0)$. We now proceed by induction.

Suppose that $\mathcal{D}_{j}\in\mathbf{O}_{\mathcal{U},u}$ with root $z_{0}$
is given, and $\mathcal{D}_{j}\in\mathbf{O}_{\mathcal{V},\langle
v(0),\dots,v(j)\rangle}$ for some $1\leq j<\operatorname{lh}(v)-1$. Let
$D_{j}\in\mathcal{D}_{j}$ be such that $D_{j}\in O(v(0),v(1),\dots,v(j))$, and
let $t_{j}\in v(j)\cap D_{j}$. In case $j=0$, make sure that
$t_{0}\not=z_{0}$. By Proposition 7.5, there is $h_{j}\in G_{v(j)\vee v(j+1)}$
which maps $t_{j}$ to some point $t_{j+1}\in v(j+1)$. Again, if $j=0$, pick
$h_{j}$ so that it also satisfies $h_{0}(z_{0})=z_{0}$. Call
$\mathcal{D}_{j+1}\coloneqq h_{j}\cdot\mathcal{D}_{j}$. Notice that the root
of $\mathcal{D}_{j+1}$ is $h_{j}(z_{0})=z_{0}$, since for $j>0$ the set $v(0)$
is pointwise fixed by $h_{j}$. Moreover, $h_{j}[D_{j}]\subseteq\bigvee_{k\leq
j+1}v(k)$, as the set on the right is setwise fixed by $h_{j}$. Lemma 7.2 then
implies

(7.3) $\mathcal{D}_{j+1}\in\mathbf{O}_{\mathcal{V},\langle
v(0),\dots,v(j+1)\rangle}.$

It is left to prove that $\mathcal{D}_{j+1}\in\mathbf{O}_{\mathcal{U},u}$. If
$\rho^{\mathcal{V}}_{\mathcal{U}}\cdot\langle
v(0),\dots,v(j+1)\rangle\sqsubseteq u$, then $\mathbf{O}_{\mathcal{V},\langle
v(0),\dots,v(j+1)\rangle}\subseteq\mathbf{O}_{\mathcal{U},u}$, in which case
we conclude by (7.3). Otherwise, it holds that $v(j)\vee
v(j+1)\subseteq\bigvee_{k<\operatorname{lh}(u)}u(k)$. By inductive assumption
$\mathcal{D}_{j}\in\mathbf{O}_{\mathcal{U},u}$, so there is
$A\in\mathcal{D}_{j}$ with $A\in O(u(0),\dots,u(-1))$. Therefore $A\cap
u(-1)\neq\emptyset$, and since $h_{j}$ fixes $u(-1)$, we can infer
$h_{j}[A]\cap u(-1)\neq\emptyset$. Moreover,
$h_{j}[A]\subseteq\bigvee_{k<\operatorname{lh}(u)}u(k)$ since the set on the
right is setwise fixed by $h_{j}$. We conclude by Lemma 7.2.

Since $g_{i},h_{j}\in G_{\mathcal{V},1}\subseteq G_{0}$ and
$\mathcal{C}_{i},\mathcal{D}_{j}\in\mathbf{O}_{\mathcal{U},u}\subseteq\mathbf{U}$
for each $i<\operatorname{lh}(P),j<\operatorname{lh}(v)$, it follows that
$\mathcal{D}_{\operatorname{lh}(v)-1}$ is in the $(\mathbf{U},G_{0})$-local
orbit of $\mathcal{C}$. But
$\mathcal{D}_{\operatorname{lh}(v)-1}\in\mathbf{O}_{\mathcal{V},v}\subseteq\mathbf{V}$
by (7.3), so we are done. ∎

We finally obtain:

######  Corollary 7.8.

Let $X$ be a strongly locally homogeneous Peano continuum which is not the
circle, the sphere, or the real projective plane. Then
$\operatorname{Homeo}(X)$ does not have the generic point property. In
particular, its universal minimal flow is not metrizable.

The same is true for any closed $G\leq\operatorname{Homeo}(X)$ which acts
locally transitively on $X$.

###### Proof.

The action $G\curvearrowright\Phi(X)$ is minimal, by Theorem 7.7. But
$\operatorname{Homeo}(X)\curvearrowright\Phi(X)$ has no comeager orbit by
Corollary 1.5, so neither does $G\curvearrowright\Phi(X)$. ∎

######  Corollary 7.9.

Let $X$ be a strongly locally homogeneous Peano continuum which is not the
circle, the sphere, or the real projective plane. Then chains on $X$ are not
classifiable by countable structures.

###### Proof.

This follows by combining Theorem 7.4, Theorem 7.7, and Corollary 1.5. ∎

######  Question 7.10.

What is the exact complexity of the orbit equivalence relation of
$\operatorname{Homeo}(X)\curvearrowright\Phi(X)$, for $X$ the torus or the
Menger curve?

######  Remark 7.11.

Corollary 7.8 was proved in the case of closed manifolds of dimension at least
3, and the Hilbert cube, in [GTZ]. Moreover, one can more easily show that
$\operatorname{Homeo}(Y)$ and the subgroup $\operatorname{Homeo}_{+}(Y)$ of
orientation preserving homeomorphism, for an orientable closed surface of
positive genus $Y$, do not have the generic point property by remarking that
such property is preserved under quotients, and that the mapping class group
$\operatorname{Homeo}_{+}(Y)/\operatorname{Homeo}_{0}(Y)$ and extended mapping
class group $\operatorname{Homeo}(Y)/\operatorname{Homeo}_{0}(Y)$ of such
surfaces are countably infinite and discrete, and thus do not have the generic
point property. This line of reasoning does not yield any information on the
dynamics of $\operatorname{Homeo}_{0}(Y)$, unlike our results.

##  Appendix A Chains on $1$-dimensional manifolds

In this appendix we prove that, in contrast with the results established in
Corollaries 7.6 and 7.9, chains on $1$-dimensional manifolds are classifiable
by countable structures and have a generic element. We prove these results for
$[0,1]$ and then reduce the case for $S^{1}$ to them. The structure of the
proof is inspired by that of [MR2156712]*Section 6.

###  A.1. Chains on $[0,1]$

We begin by giving a more concrete description of $\Phi([0,1])$ which we will
then use for the whole section. Recall from Section 2.1, that we denote by
$C([0,1])$ the space of subcontinua of $[0,1]$. The elements of $C([0,1])$ are
closed intervals $[x,y]$ with $0\leq x\leq y\leq 1$ and the natural map
$\varphi\colon C([0,1])\to\mathbb{R}^{2}$ given by $\varphi([x,y])=(x,y)$ is a
homeomorphism onto its image, which is the closed triangle $T$ with vertices
$(0,0)$, $(0,1)$ and $(1,1)$ [illanes1999hyperspaces]*Example 5.1. Elements of
$C([0,1])$ are partially ordered by inclusion. For pairs of points in $T$,
this translates to the partial order $(x,y)\sqsubseteq(x^{\prime},y^{\prime})$
iff $x\geq x^{\prime}$ and $y\leq y^{\prime}$.

Recall that every chain in $\Phi([0,1])$ is the image of an injective
continuous order-preserving function
$c\colon([0,1],\leq)\to(C([0,1]),\subseteq)$ such that $c(0)$ is a singleton
and $c(1)=[0,1]$. Under the identification of $C([0,1])$ with $T$ described
above, every chain $\mathcal{C}\in\Phi([0,1])$ gives rise to an arc
$\varphi\cdot\mathcal{C}\subseteq T$ containing the vertex $(0,1)$ and a point
$(x_{0},x_{0})$, for some $x_{0}\in[0,1]$, on the diagonal, and such that for
all $(x,y),(x^{\prime},y^{\prime})\in\varphi\cdot\mathcal{C}$, either
$(x,y)\sqsubseteq(x^{\prime},y^{\prime})$ or
$(x^{\prime},y^{\prime})\sqsubseteq(x,y)$. In the following we will identify
$\Phi([0,1])$ with the subspace of $C(T)$ of such arcs, dropping any mention
of $\varphi$.

$\operatorname{Homeo}_{+}([0,1])$ acts on $T$ diagonally, that is,
$h\cdot(x,y)=(h(x),h(y))$. The induced action on $\Phi([0,1])\subseteq C(T)$
coincides with the natural action
$\operatorname{Homeo}_{+}([0,1])\curvearrowright\Phi([0,1])$, discussed in
Section 2.

######  Definition A.1.

Given a chain $\mathcal{C}\subseteq T$, a closed subset
$\mathcal{I}\subseteq\mathcal{C}$ is called a _subchain (of $\mathcal{C}$)_ if
it is connected (in $T$). A non-trivial subchain $\mathcal{I}$ is

  1. (1)

_horizontal_ if for all $(x,y),(x^{\prime},y^{\prime})\in\mathcal{I}$, we have
$y=y^{\prime}$,

  2. (2)

_vertical_ if for all $(x,y),(x^{\prime},y^{\prime})\in\mathcal{I}$, we have
$x=x^{\prime}$,

  3. (3)

_strictly monotone_ in $\mathcal{C}$ if for all
$(x,y),(x^{\prime},y^{\prime})\in\mathcal{I}$, we have $x=x^{\prime}$ iff
$y=y^{\prime}$.

A subchain $\mathcal{I}\subseteq\mathcal{C}$ is _maximal horizontal_ if it is
horizontal and it is not properly contained in any horizontal subchain of
$\mathcal{C}$. Analogously, we define _maximal vertical subchains_ and
_maximal strictly monotone subchains_. By Zorn’s Lemma, each
horizontal/vertical/strictly monotone subchain, is contained in a maximal
subchain of the same type. A subchain is _maximal_ if it is either maximal
horizontal, maximal vertical, or maximal strictly monotone.

Each subchain $\mathcal{I}\subseteq\mathcal{C}$ has a maximum and a minimum
element with respect to $\sqsubseteq$. If $(x_{1},y_{1}),(x_{2},y_{2})$ are
the maximum and minimum of $\mathcal{I}$, we call the intervals
$[x_{1},x_{2}]$ and $[y_{2},y_{1}]$ in $[0,1]$ the _domain_ and _codomain_ of
$\mathcal{I}$, respectively, and denote them by $\mathrm{dom}(\mathcal{I})$
and $\mathrm{codom}(\mathcal{I})$, see Figure 2. Notice that if $\mathcal{I}$
is maximal, then at least one between the domain and codomain is non-trivial.

[scale=1]img/domain

Figure 2. Domain and codomain of a subchain $\mathcal{I}$.

Note that if $\mathcal{C}\subseteq T$ is a chain, and
$\mathcal{I},\mathcal{I}^{\prime}$ are distinct maximal subchains, then their
domains meet in at most one point, and the same is true for their codomains.
Since the domains of maximal horizontal and maximal strictly monotone
subchains are non-trivial, it follows that there are only countably many such
subchains. By reasoning about codomains, we also see that there are countably
many maximal vertical subchains, so that $\mathcal{C}$ contains countably many
maximal subchains.

We claim that if $(x_{0},x_{0})$ is the root of $\mathcal{C}$, then the union
of the domains of maximal subchains of $\mathcal{C}$ is dense in $[0,x_{0}]$,
while the union of their codomains is dense in $[x_{0},1]$. The projection of
$\mathcal{C}$ on the $x$-axis is $[0,x_{0}]$ and on the $y$-axis it is
$[x_{0},1]$. Suppose $x_{0}\neq 0,1$, as otherwise the statement is trivial.
Let us show the result for the domains, the result for the codomains follows
analogously. Fix $0\leq x_{1}\leq x_{2}\leq x_{0}$, and let
$(x_{1},y_{1}),(x_{2},y_{2})$ be points in $\mathcal{C}$. If the subchain
$\mathcal{I}$ with maximum $(x_{1},y_{1})$ and minimum $(x_{2},y_{2})$ is not
strictly monotone, there is a subchain included in it which is either vertical
or horizontal. In all three cases, $\mathcal{I}$ intersects a maximal subchain
whose domain contains a point between $x_{1}$ and $x_{2}$, and we are done.

Before proceeding with the proof of the classifiability result we are after,
we need to check that a few sets that will arise in its proof are Borel, which
we do in the following two lemmas.

######  Lemma A.2.

The following subsets of $\Phi([0,1])\times C([0,1])$ (identified with a
subspace of $C(T)\times T$ as above) are Borel:

$\displaystyle\mathcal{E}^{H}$