# Continuous Lebesgue measure-preserving maps on one-dimensional manifolds: a
survey

Jozef Bobok  ,  Jernej Činč  ,  Piotr Oprocha  and  Serge Troubetzkoy
Department of Mathematics of FCE, Czech Technical University in Prague,
Thákurova 7, 166 29 Prague 6, Czech Republic<EMAIL_ADDRESS>Centre of
Excellence IT4Innovations - Institute for Research and Applications of Fuzzy
Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1, Czech
Republic<EMAIL_ADDRESS>Centre of Excellence IT4Innovations - Institute
for Research and Applications of Fuzzy Modeling, University of Ostrava, 30.
dubna 22, 701 03 Ostrava 1, Czech Republic<EMAIL_ADDRESS>Aix Marseille
Univ, CNRS, Centrale Marseille, I2M, Marseille, France postal address: I2M,
Luminy, Case 907, F-13288 Marseille Cedex 9, France  serge.troubetzkoy@univ-
amu.fr

###### Abstract.

We survey the current state-of-the-art about the dynamical behavior of
continuous Lebesgue measure-preserving maps on one-dimensional manifolds.

###### Key words and phrases:

Interval map, circle map, Lebesgue measure-preserving, periodic points,
generic properties

###### 2020 Mathematics Subject Classification:

37E05, 37A05, 37B65, 37C20

##  1\. Introduction

Let $M$ denote a compact connected one-dimensional manifold, namely the unit
interval, $I:=[0,1]$, and the unit circle, $\mathbb{S}^{1}$. Define $C(M)$ to
be the set of continuous maps of $M$. Let $\lambda$ denote the Lebesgue
measure on an underlying manifold. Our survey will focus on discussion of
topological and dynamical properties in the space $C_{\lambda}(M)\subset C(M)$
of Lebesgue measure-preserving continuous maps of $M$ with the metric of
uniform convergence; the space $C_{\lambda}(M)$ is a complete metric space,
see [12, Proposition 4]. Our particular interest is in Lebesgue measure-
preserving continuous maps on $M$ which are not necessarily invertible, so the
case that is not usually studied in Ergodic Theory.

The class $C_{\lambda}(M)$ contains very large spectrum of maps; on one hand
nowhere differentiable ones or even without finite or infinite one-sided
derivative [10] and on the other hand many piecewise monotone maps, many
piecewise smooth maps and of course maps id and $1-\textrm{id}$. Furthermore,
$C_{\lambda}(\mathbb{S}^{1})$ also contains all circle rotations.

For a general compact manifold $\mathbb{M}$, let $H_{\lambda}(\mathbb{M})$
denote the space of Lebesgue measure-preserving homeomorphisms of
$\mathbb{M}$, which is again a complete metric space when equipped with the
uniform metric. In the setting of volume preserving homeomorphisms in
dimension 1, the dynamical behavior is simple and thus not of much interest.
However, there are some similarities of the dynamics of higher dimensional
homeomorphisms with the one dimensional continuous maps, so we will mention
them throughout the article. There is a survey book by Alpern and Prasad on
the dynamics of generic volume preserving homeomorphism [4], thus we only
briefly mention some such results for comparison with $C_{\lambda}(M)$.

Our choice of $C_{\lambda}(M)$ (and $H_{\lambda}(\mathbb{M})$) is motivated by
the fact that they are one-dimensional versions of volume-preserving maps, or
more broadly, conservative dynamical systems; ergodic maps preserving Lebesgue
measure are the most fundamental examples of maps having a unique physical
measure. Since generic maps in $C_{\lambda}(M)$ are weakly mixing [17], the
Ergodic Theorem implies that for a generic map in $C_{\lambda}(M)$ the closure
of a typical trajectory has full Lebesgue measure, thus the statistical
properties of typical trajectories can be revealed by physical observations.

Or as Karl Petersen says in the introduction of [47]:

> Measure-preserving systems model processes in equilibrium by transformations
> on probability spaces or, more generally, measure spaces. They are the basic
> objects of study in ergodic theory, a central part of dynamical systems
> theory. These systems arise from science and technology as well as from
> mathematics itself, so applications are found in a wide range of areas, such
> as statistical physics, information theory, celestial mechanics, number
> theory, population dynamics, economics, and biology.

On the other hand, they represent a variety of possible one-dimensional
dynamics as highlighted in the following remark which is proven in the
interval case in the introduction of [12]; in the circle case the proof is
analogous using Lemma A.1 from the Appendix.

######  Remark 1.1.

Let $f\in C_{\lambda}(M)$, then (i) and (ii) are equivalent, and (iii) implies
(i).

  * (i)

$f$ preserves a non-atomic probability measure $\mu$ with
$\mathrm{supp}~{}\mu=M$.

  * (ii)

There exists a homeomorphism $h$ of $M$ such that $h\circ f\circ h^{-1}\in
C_{\lambda}(M)$.

  * (iii)

$f$ has a dense set of periodic points, i.e., $\overline{\mathrm{Per}(f)}=M$.

Furthermore, if $\mathrm{Per}(f)\neq\emptyset$, we have (i) implies (iii).
Otherwise $f$ is conjugate to an irrational rotation.

The main line of research in this area has been describing the size of the set
of maps satisfying a certain topological or dynamical property. These results
are discussed in Section 2 and are summarized in the following table.

Property | $C_{\lambda}{(M)}$ | Result
---|---|---
Weak mixing | generic | Theorem 2.2.1 & 2.2.3
Strong mixing | dense, but of first category | Theorem 2.2.2
Leo | open dense | Theorem 2.3.2 & 2.3.4
Shadowing | generic | Corollary 2.5.2 & Theorem 2.5.4
Periodic shadowing | generic | Theorem 2.5.2 & Corollary 2.5.5
Specification prop. | open dense | Corollary 2.3.3 & 2.3.5
Hausdorff dim. $\Gamma_{f}$ | 1 | Theorem 2.4.1
Lower box dim. $\Gamma_{f}$ | 1 | Theorem 2.4.1
Upper box dim. $\Gamma_{f}$ | 2 | Theorem 2.4.1
$\lambda$-a.e. knot point | generic | Theorem 2.6.1
$\delta$-crookedness | generic | Theorem 2.7.2

While the previous table summarizes the properties which are the same in
$C_{\lambda}(I)$ and $C_{\lambda}(\mathbb{S}^{1})$ there are some properties
that are different or we do not known if they are the same. In particular,
Theorem 2.5.4 shows that s-limit shadowing is generic in
$C_{\lambda}(\mathbb{S}^{1})$, however, we can only prove the s-limit
shadowing property and limit shadowing property are dense in $C_{\lambda}(I)$
(see Proposition 2.5.3). We also give a theorem describing the structure and
dimensions of the set of periodic points of generic maps from $C_{\lambda}(I)$
and $C_{\lambda}(\mathbb{S}^{1})$. Results are identical in these settings
(see Theorems 2.4.2 and 2.4.3) except for the circle maps of degree one (see
Theorem 2.4.4).

In Section 3 we state Theorems 3.1.1 and 3.1.2 that show that certain
topological and metric properties are equivalent for sufficiently smooth maps.
In Subsection 3.2 we state Theorem 3.2.1 which shows that there exists a
nowhere monotone map in $C_{\lambda}(I)$ that has finite topological entropy.
This motivates a more general interesting question regarding the connection
between nowhere differentiability and infinite topological entropy.

We finish the article with Section 4 where we give an overview of known
related results for spaces of maps on $M$ equipped with smoother topologies.

Not to disturb the flow of reading, we give in the appendix a proof of Lemma
Lemma A.1 which is needed to argue for Remark 1.1.

##  2\. Denseness properties

First one needs to note that both spaces $C_{\lambda}(I)$ and
$C_{\lambda}(\mathbb{S}^{1})$ equipped with the metric of uniform convergence
are complete. This follows from Proposition 4 of [12] which is proven for
$M=I$, however the analogous proof works also for $\mathbb{S}^{1}$. We call a
dense $G_{\delta}$ set residual and call a property generic if it is attained
on at least a residual set of the Baire space on which we work. In this
section we will study three different notions of denseness. The strongest
property that we verify is that a certain dynamical property holds for an open
dense set of maps in $C_{\lambda}(M)$. We verify the weaker notion of
genericity for many other topological dynamical properties, while for certain
properties we verify only the denseness of maps with certain properties in
$C_{\lambda}(M)$.

The study of generic properties in dynamical systems was initiated in the
article by Oxtoby and Ulam from 1941 [46] in which they showed that for a
finite-dimensional compact manifold with a non-atomic measure that is positive
on open sets, the set of ergodic measure-preserving homeomorphisms is a
generic set in the strong topology. Later Halmos in 1944 [32],[33] introduced
approximation techniques to a purely metric setting. Namely, he studied
interval maps which are invertible almost everywhere and preserve the Lebesgue
measure. He showed that the generic invertible map is weakly mixing (i.e., has
continuous spectrum). Subsequently, Rohlin in 1948 [50] showed that Halmos’
result is optimal in a sense that the set of strongly mixing measure-
preserving invertible maps is of the first category in the same underlying
space. It took until 1967 that this line of research was continued when Katok
and Stepin [37] introduced the notion of a speed of approximation. One of the
most notable applications of their methods is the proof of genericity of
ergodicity and weak mixing for certain classes of interval exchange
transformations (IETs). More details on the follow-up history of approximation
theory can be found in the surveys [26], [5] and [54].

Now we restrict to our particular context. The roots for studying generic
properties on $C_{\lambda}(I)$ come from the paper [10] and this line of study
was continued recently in [17], [11], [12], [13]. The first observation we can
make about maps from $C_{\lambda}(I)$ is that they have dense set of periodic
points. This follows directly from the Poincaré Recurrence Theorem and the
fact that in dynamical systems given by an interval map the closures of
recurrent points and periodic points coincide [25]. Furthermore, except for
the two exceptional maps $\mathrm{id}$ and $1-\mathrm{id}$, every such map has
positive metric entropy. In fact, except for these two exceptional maps every
map is non-invertible on a set of positive measure and thus by a well known
theorem (see for example [55, Corollary 4.14.3]) has positive metric entropy
and thus positive topological entropy as well.

###  2.1. Main tools

There are two main tools that are used in most of the results from this
section.

Building on the work of Bobok [10, Lemma 1] the following lemma was proven in
[17, Proposition 12] for the interval case, for the definition of a leo map we
refer the reader to Subsection 2.3.

######  Lemma 2.1.1.

The set of maps that are piecewise affine Markov and leo are dense in
$C_{\lambda}(M)$.

###### Proof.

The proof in the circle case follows by combining various known results. First
of all the density of a special collection of maps in
$PA_{\lambda}(\mathbb{S}^{1})$ was shown in Lemmas 13 and 14 from [11], we can
assume all of the absolute values of slopes of these maps are at least 4. Then
using Lemma 12 from [13] we can find a dense set of maps in
$PA_{\lambda}(\mathbb{S}^{1})$ all of whose critical values are distinct and
again of whose slopes are at least 4. Finally using Lemma 12 from [17] whose
proof also works without change for the circle we have that there is a dense
set of Markov maps in $PA_{\lambda}(\mathbb{S}^{1})$. In this proof it is
implicitly left as an exercise to the reader to show that the resulting map is
leo, which we do here. Using the notation of the proof of this lemma in [17],
let $J:=(p_{i-1},p_{i+1})$ and $U$ an arbitrary non-empty open interval. Since
the slopes of $f$ and $g_{1}$ are at least 4 we have
$\lambda(g_{1}(A))/\lambda(A)\geq 4/2=2$ for any non-empty interval which
contains at most one critical point, thus we can find an $n$ such that
$V:=g_{1}^{n}(V)$ contains at least two consecutive critical points. From the
construction of $g_{1}$ we have $f(V\setminus J)=g_{1}(V\setminus J)$,
$f(V\cap J)\subset g_{1}(V\cap J)$ provided that
$\\{p_{i-1},p_{i+1}\\}\neq\emptyset$, and if $V\subset J$ then since $V$
contains two consecutive critical points we have
$g_{1}(V)=g_{1}(J)=f(J)\supset f(V)$; and thus $g_{1}$ is leo. ∎

The proofs in this section also use window perturbations as the other main
tool. Let $J$ be an arc in $M$ (i.e., a homeomorphic image of $[0,1]$). Let
$m$ be an odd positive integer and $\\{J_{i}\in M:1\leq i\leq m\\}$ a finite
collection of arcs satisfying $\cup^{m}_{i=1}J_{i}=J$ and
$\operatorname{int}(J_{i})\cap\operatorname{int}(J_{j})=\emptyset$ when $i\neq
j$. We will refer to this as a partition of $J$.

Fix $f\in C_{\lambda}(M)$ an arc $J\in M$ and a partition $\\{J_{i}\\}$ of
$J$. A map $h\in C(M)$ is an $m$-fold window perturbation of $f$ with respect
to $J$ and the partition $\\{J_{i}\\}$ if

  * •

$h|_{J^{c}}=f|_{J^{c}}$

  * •

for each $1\leq i\leq m$ the map $h|_{J_{i}}$ is an affinely scaled copy of
$f|_{J}$ with the orientation reversed for every second $i$, with $h|_{J_{1}}$
having the same orientation as $f|_{J}$.

The essence of this definition is illustrated by Figure 1.

$f$$a$$b$$0$$1$$1$

$h$$0$$1$$1$$a$$b$

Figure 1. For $f\in C_{\lambda}(M)$ shown on the left, we show on the right
the graph of $h$ which is a $3$-fold piecewise window perturbation of $f$ on
the interval $[a,b]$.

###  2.2. Measure-theoretic properties

Let $\mathcal{B}$ denote the Borel sets in $M$. The measure-preserving
transformation $(f,M,\mathcal{B},\mu)$ is called

  * •

ergodic if for every $A\in\mathcal{B}$, $f^{-1}(A)=A$ $\mu$-a.e. implies that
$\mu(A)=0$ or $\mu(A^{c})=0$.

  * •

weakly mixing, if for every $A,B\in\mathcal{B}$,

$\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}|\mu(f^{-j}(A)\cap
B)-\mu(A)\mu(B)|=0.$

  * •

strongly mixing if for every
$A,B\in\mathcal{B},\lim_{n\to\infty}\mu(f^{j}(A)\cap B)=\mu(A)\mu(B)$.

  * •

measure-theoretically exact if for each set $A\in\cap_{n\geq
0}f^{-n}(\mathcal{B})$ it holds that $\mu(A)\mu(A^{c})=0$.

In this paper we will mainly focus on the case of particular invariant
measure, the Lebesgue measure $\lambda$.

The following measure-theoretic properties of $C_{\lambda}(I)$-generic
functions were proven in [17].

######  Theorem 2.2.1.

The $C_{\lambda}(I)$-generic function

  1. (1)

is weakly mixing with respect to Lebesgue measure $\lambda$ [17, Th. 15],

  2. (2)

maps a set of Lebesgue measure zero onto $I$ [17, Cor. 22].

In analogy to Rohlin’s classical result [50], the following result was proven
for the interval in [17, §3]. For the circle case we need several
modifications, first for the density [17, Proposition 12] is replaced by Lemma
2.1.1. Except for the application of the Block Coven result ([6, Theorem 9],
[17, Theorem 18]), all the other steps work for the circle without change. To
apply [6, Theorem 9] in the circle case it suffices to start with a
sufficiently fine Markov partition such that the diameter of the image of each
element of the partition is less than the diameter of the circle.

######  Theorem 2.2.2.

The set of mixing maps in $C_{\lambda}(M)$ is dense and is of the first
category.

Returning to the case when $M=\mathbb{S}^{1}$, the following result was
obtained in [13, Theorem 2]. For each $\alpha\in[0,1)]$, let
$r_{\alpha}\colon\mathbb{S}^{1}\to\mathbb{S}^{1}$ be a circle rotation for the
angle $\alpha$. Define the operator $T_{\alpha,\beta}\colon
C_{\lambda}(\mathbb{S}^{1})\mapsto C_{\lambda}(\mathbb{S}^{1})$ by
$T_{\alpha,\beta}(f)=r_{\alpha}\circ f\circ r_{\beta}.$

######  Theorem 2.2.3.

There exists a dense $G_{\delta}$ subset $\mathcal{G}$ of
$C_{\lambda}(\mathbb{S}^{1})$ such that

  1. (1)

each $g\in\mathcal{G}$ is weakly mixing with respect to $\lambda$,

  2. (2)

each $g\in\mathcal{G}$ maps a set of Lebesgue measure zero onto
$\mathbb{S}^{1}$, and

  3. (3)

for each pair $\alpha,\beta\in[0,1)$ and each $g\in\mathcal{G}$ the map
$T_{\alpha,\beta}(g)\in\mathcal{G}$.

Point (2) was shown in [17, Cor. 22] for the interval, the proof holds without
change in the case of the circle, and furthermore if $g$ maps a set of
Lebesgue measure zero onto $\mathbb{S}^{1}$, then so does
$T_{\alpha,\beta}(g)$.

In [12, Theorem 2] it was shown that there is a dense set of non-ergodic maps
in $C_{\lambda}(I)$; its proof works with natural modifications also in
$C_{\lambda}(\mathbb{S}^{1})$. Thus Theorem 2.2.3 is optimal since there is no
nonempty open set of maps satisfying nice mixing properties for any
$\alpha,\beta$.

For volume preserving homeomorphisms we have the following classical result on
the unit cube $I^{n}$ due to Oxtoby and Ulam [46] (see also[4, Theorem 7.1]):

######  Theorem 2.2.4.

The generic $f\in H_{\lambda}(I^{n})$ is ergodic for all $n\geq 2$.

The improvement of Theorem 2.2.4 was shown in [2, Theorem 1.2]:

######  Theorem 2.2.5.

The generic $f\in H_{\lambda}(\mathbb{M})$ is weakly mixing and periodic
points of $f$ are dense in $\mathbb{M}$ for any compact manifold $\mathbb{M}$
of dimension at least 2.

###  2.3. Topological expansion properties

We call a map $f\in C_{\lambda}(M)$

  * •

transitive if for all nonempty open sets $U,V\subset M$ there exists $n\geq 0$
so that $f^{n}(U)\cap V\neq\emptyset$,

  * •

totally transitive if $f^{n}$ is transitive for all $n\geq 1$.

  * •

(topologically) weakly mixing if its Cartesian product $f\times f$ is
transitive,

  * •

topologically mixing if for all nonempty open sets $U,V\subset M$ there exists
$n_{0}\geq 0$ so that $f^{n}(U)\cap V\neq\emptyset$ for every $n\geq n_{0}$,

  * •

leo (locally eventually onto, also known as topologically exact) if for every
nonempty open set $U\subset M$ there exists $n\in{\mathbb{N}}$ so that
$f^{n}(U)=M$.

By the usual hierarchy in topological dynamics, every leo map is topologically
mixing, topologically weakly mixing, totally transitive and transitive. In
[17, Theorem 9] the following theorem was shown:

######  Theorem 2.3.1.

The $C_{\lambda}(I)$-generic function is leo.

Theorem 2.3.1 was recently improved in [14] where the following result was
shown:

######  Theorem 2.3.2.

There is an open dense set of leo maps in $C_{\lambda}(I)$.

If we take any continuous map of $I$ with attracting fixed point in the
interior of $I$ then it is clear that all sufficiently close maps in $C(I)$
cannot be transitive. For any map in $C(I)$ we can blow up an attracting to an
invariant set with attracting fixed point. Thus non-transitive continuous
interval maps and also continuous interval maps with
$\overline{\mathrm{Per}(f)}\neq I$ form open dense sets.

By the result of Blokh [9, Theorem 8.7], for interval maps topological mixing
implies periodic specification property (cf. proof of Buzzi [20, Appendix A]).
Therefore, we obtain the following corollary.

######  Corollary 2.3.3.

There is an open dense set of maps from $C_{\lambda}(I)$ satisfying the
periodic specification property.

Now we turn to the the circle case, where a more detailed study has been
realized recently in [13].

######  Theorem 2.3.4.

There is an open dense set of maps $\mathcal{O}\subset
C_{\lambda}(\mathbb{S}^{1})$ such that:

  1. (1)

each map $f\in\mathcal{O}$ is leo.

  2. (2)

for each pair of $\alpha,\beta\in[0,1)$ and each $f\in\mathcal{O}$ the map
$T_{\alpha,\beta}(f)\in\mathcal{O}$.

In fact, Blokh’s results about specification property mentioned above
generalize to all topologically mixing maps on topological graphs, in
particular to the circle (see [7], cf. [8, 3]; another approach for proving
the generalization of Blokh’s results on topological graphs can be found in
[34] and was inspired by the techniques of Buzzi for interval maps [20,
Appendix A].

######  Corollary 2.3.5.

There is an open dense set of maps from $C_{\lambda}(\mathbb{S}^{1})$
satisfying the periodic specification property.

By the same argument as for the interval maps, this result does not extend to
the whole set $C(\mathbb{S}^{1})$.

We turn to the case of Lebesgue measure-preserving homeomorphisms. A map
$f:X\to X$ of a compact metric space $X$ is called maximally chaotic if

  1. (1)

$f$ is topologically transitive,

  2. (2)

the periodic points of $f$ are dense in $X$, and

  3. (3)

$\limsup_{k\to\infty}\text{diam}(f^{k}(U))=\text{diam}(X)$ for any non-empty
open set $U\subset X$.

Notice that maximal chaos implies the well known Devaney chaos. While the leo
property is impossible for homeomorphisms, we have the following result in the
setting of the $n$-dimensional cube $I^{n}$, which summarizes the results
found in [4, Theorems 4.5 and D]:

######  Theorem 2.3.6.

For $n\geq 2$ the generic $f\in H_{\lambda}(I^{n})$ is topologically weakly
mixing and maximally chaotic.

It is not hard to see that conditions (1) and (3) in the definition of maximal
chaos are immediate consequence of weak mixing. Condition (2) is not in
general consequence of weak mixing, since there exist weakly mixing minimal
systems.

###  2.4. Periodic points and dimension properties

The Hausdorff, lower and upper box dimension of the graph of a function is a
way to describe the “roughness” of the function. The following theorem by
Schmeling and Winkler [53] was stated for maps from $C_{\lambda}(I)$, but it
holds in any dimension111Personal communication from Jörg Schmeling.; in
particular, in dimension one we have the following theorem.

######  Theorem 2.4.1.

A graph of a generic map $f\in C_{\lambda}(M)$ has Hausdorff dimension equal
to the lower box dimension, both equal to $1$ and the upper box dimension
equal to $2$.

Understanding of the structure of the set of periodic points of the function
under consideration is among the fundamental tasks in dynamical systems
theory. Since generic maps from $C_{\lambda}(I)$ are weakly mixing with
respect to $\lambda$ it follows that the Lebesgue measure of the set of
periodic points is equal to $0$. It is interesting to study the finer
structure of the set of periodic points for generic maps, in particular its
cardinality and dimension.

The set of periodic points of period $k$ for $f$ is denoted
$\mathrm{Per}(f,k)$, the set of fixed points of $f^{k}$ is denoted by
$\mathrm{Fix}(f,k)$ and of the union of all periodic points of $f$ is denoted
by $\mathrm{Per}(f)$. In [12] the authors studied the cardinality and
structure of the set of periodic points and its respective lower box, upper
box and Hausdorff dimensions:

######  Theorem 2.4.2.

For a generic map $f\in C_{\lambda}(I)$, for every $k\geq 1$:

  1. (1)

$\mathrm{Fix}(f,k)$ is a Cantor set,

  2. (2)

$\mathrm{Per}(f,k)$ is a relatively open dense subset of the set
$\mathrm{Fix}(f,k)$,

  3. (3)

the set $\mathrm{Fix}(f,k)$ has lower box dimension and Hausdorff dimension
zero. In particular, $\mathrm{Per}(f,k)$ has lower box dimension and Hausdorff
dimension zero. As a consequence, the Hausdorff dimension of $\mathrm{Per}(f)$
is also zero.

  4. (4)

the set $\mathrm{Per}(f,k)$ has upper box dimension one. Therefore,
$\mathrm{Fix}(f,k)$ has upper box dimension one as well.

In the setting of $C_{\lambda}(\mathbb{S}^{1})$, due to the presence of
rotations, degree $1$ maps need to be treated separately. Denote the set of
degree $d$ maps in $C_{\lambda}(\mathbb{S}^{1})$ by
$C_{\lambda,d}(\mathbb{S}^{1})$.

The proof of Theorem 2.4.2 shows:

######  Theorem 2.4.3.

Conclusions of Theorem 2.4.2 hold also for generic maps in
$C_{\lambda,d}(\mathbb{S}^{1})$ for each $d\in\mathbb{Z}\setminus\\{1\\}$.

The proofs of these two theorems can easily be adapted to show that the
generic map in $C(I)$ and degree $d$ maps in $C(\mathbb{S}^{1})$ (i.e., not
necessarily measure preserving) have the same properties (see [12][Remark 14
and 18]).

For $C_{\lambda,1}(\mathbb{S}^{1})$ the situation is more complicated. A
periodic point $x\in\mathrm{Per}(f,k)$ is called _transverse_ , if the graph
of $f^{k}$ crosses the diagonal at $x$ (possibly coincides with the diagonal
on an interval containing $x$). Consider the open set

$C_{p}:=\\{f\in C_{\lambda,1}(\mathbb{S}^{1}):f\text{ has a transverse
periodic point of period }p\\}.$

In this setting the proof of Theorem 2.4.2 yields a similar result from [12].

######  Theorem 2.4.4.

For any $f$ in a dense $G_{\delta}$ subset of $\overline{C}_{p}$ we have for
each $k\in\mathbb{N}$

  1. (1)

$\mathrm{Fix}(f,kp)$ is a Cantor set,

  2. (2)

$\mathrm{Per}(f,kp)$ is a relatively open dense subset of
$\mathrm{Fix}(f,kp)$,

  3. (3)

$\mathrm{Fix}(f,kp)$ has lower box dimension and Hausdorff dimension zero. In
particular, $\mathrm{Per}(f,kp)$ has lower box dimension and Hausdorff
dimension zero. As a consequence, the Hausdorff dimension of $\mathrm{Per}(f)$
is zero as well.

  4. (4)

the set $\mathrm{Per}(f,kp)$ has upper box dimension $1$. Thus,
$\mathrm{Fix}(f,kp)$ and $\mathrm{Per}(f)$ both have upper box dimension also
$1$.

To interpret this result the set
$C_{\infty}:=C_{\lambda,1}(\mathbb{S}^{1})\setminus\cup_{p\geq
1}\overline{C_{p}}$ was studied in [12]. It turns out that a periodic point
can be transformed to a transverse periodic point by an arbitrarily small
perturbation of the map, thus the set $C_{\infty}$ consists of maps without
periodic points. Using the same argument one can see that $\cup_{p\geq
1}\overline{C_{p}}$ contains an open dense set. Therefore, $C_{\infty}$ is
nowhere dense in $C_{\lambda,1}(\mathbb{S}^{1})$. Furthermore, in [12] the
following complete characterization of the set $C_{\infty}$ was obtained:

######  Proposition 2.4.5.

The set $C_{\infty}$ consists of irrational circle rotations.

Periodic points of generic homeomorphisms have been well studied also in other
contexts. Akin et. al. proved that the set of periodic points is a Cantor set
for generic homeomorphisms of $\mathbb{S}^{1}$ [1, Theorems 9.1 and 9.2(a)].
On the other hand, Carvalho et. al. have shown that the upper box dimension of
the set of periodic points is full (i.e., of the same dimension as the
dimension of the underlying manifold) for generic homeomorphisms on compact
manifolds of dimension at least one [21]222this statement only appears in the
published version of [21]..

Let us recall that generic maps from $C_{\lambda}(I)$ will necessarily have
Lebesgue measure $0$ on the set of periodic points since $\lambda$ is weakly
mixing. Nonetheless the following somewhat surprising result is proven in [12,
Theorem 2].

######  Theorem 2.4.6.

The set of leo maps in $C_{\lambda}(I)$ whose periodic points have full
Lebesgue measure and whose periodic points of period $k$ have positive
Lebesgue measure for each $k\geq 1$ is dense in the set $C_{\lambda}(I)$.

The following result about volume preserving homeomorphisms is proven in an
unpublished sketch by Guihéneuf [30].

######  Theorem 2.4.7.

The set of periodic points of a generic $f\in H_{\lambda}(\mathbb{M})$ for a
compact manifold $\mathbb{M}$ of dimension at least two is a dense set of zero
measure and for every $\ell\geq 1$ the set of fixed points of $f^{\ell}$ is
either empty or perfect.

More generally Guihéneuf’s result holds for homeomorphisms preserving a “good”
measure in the sense of Oxtoby and Ulam [46].

###  2.5. Shadowing properties

One of the classical notions from topological dynamics is the so-called
shadowing property. It is of particular importance in systems possessing
sensitive dependence on initial conditions. In such systems, very small errors
could potentially lead to large divergence of orbits. Shadowing is a notion
arising from computer science and is used as a tool for determining if any
hypothetical orbit is indeed close to some real orbit of a topological
dynamical system. It assures that the dynamics of maps which satisfy it can be
realistically observed through computer simulations. Let us first give
definitions that are important for this subsection.

For $\delta>0$, we call a sequence $(x_{n})_{n\in\mathbb{N}_{0}}\subset I$ a
_$\delta$ -pseudo orbit_ of $f\in C(I)$ if $d(f(x_{n}),x_{n+1})<\delta$ for
every $n\in\mathbb{N}_{0}$. A _periodic $\delta$-pseudo orbit_ is a
$\delta$-pseudo orbit for which there is $N\in\mathbb{N}_{0}$ so that
$x_{n+N}=x_{n}$, for every $n\in\mathbb{N}_{0}$. The sequence
$(x_{n})_{n\in\mathbb{N}_{0}}$ is called an _asymptotic pseudo orbit_ if
$\lim_{n\to\infty}d(f(x_{n}),x_{n+1})=0$. Provided a sequence
$(x_{n})_{n\in\mathbb{N}_{0}}$ is a $\delta$-pseudo orbit and an asymptotic
pseudo orbit we say it is an asymptotic $\delta$-pseudo orbit.

######  Definition 2.5.1.

We say that a map $f\in C(M)$ has the:

  * •

_shadowing property_ if for every $\varepsilon>0$ there exists $\delta>0$
satisfying the following: given any $\delta$-pseudo orbit
$\mathbf{y}:=(y_{n})_{n\in\mathbb{N}_{0}}$ we can find a corresponding point
$x\in M$ that $\varepsilon$-traces $\mathbf{y}$, i.e.,

$d(f^{n}(x),y_{n})<\varepsilon\text{ for every }n\in\mathbb{N}_{0}.$

  * •

_periodic shadowing property_ if for every $\varepsilon>0$ there exists
$\delta>0$ satisfying the following condition: given any periodic
$\delta$-pseudo orbit $\mathbf{y}:=(y_{n})_{n\in\mathbb{N}_{0}}$ we can find a
corresponding periodic point $x\in M$, which $\varepsilon$-traces
$\mathbf{y}$.

  * •

_limit shadowing property_ if for every _asymptotic pseudo-orbit_ , i.e.
sequence $(x_{n})_{n\in\mathbb{N}_{0}}\subset M$, so that

$d(f(x_{n}),x_{n+1})\to 0\text{ when }n\to\infty$

there exists $p\in M$, so that

$d(f^{n}(p),x_{n})\to 0\text{ as }n\to\infty.$

  * •

_s-limit shadowing property_ if for every $\varepsilon>0$ there exists
$\delta>0$ so that

    1. (1)

for every $\delta$-pseudo orbit $\mathbf{y}:=(y_{n})_{n\in\mathbb{N}_{0}}$ we
can find a corresponding point $x\in M$ which $\varepsilon$-traces
$\mathbf{y}$,

    2. (2)

for every asymptotic $\delta$-pseudo orbit
$\mathbf{y}:=(y_{n})_{n\in\mathbb{N}_{0}}$ of $f$, there exists $x\in M$ which
$\varepsilon$-traces $\mathbf{y}$ and

$\lim_{n\to\infty}d(y_{n},f^{n}(x))=0.$

The following theorem was proved by the authors in [12, Theorem 3].

######  Theorem 2.5.2.

The shadowing and periodic shadowing properties are generic for maps from
$C_{\lambda}(I)$.

For comparison, in the larger space $C(M)$ Mizera proved in [42] that the
shadowing property is generic. Several other results that shadowing is generic
in topology of uniform convergence in more general settings were established
(see [43, 38]) using the techniques of Pilyugin and Plamenevskaya [48]
initially developed for proving genericity of shadowing property of
homeomorphisms on any smooth compact manifolds without a boundary.

######  Proposition 2.5.3.

The set of maps $f\in C_{\lambda}(I)$ that have s-limit shadowing property is
dense in the set $C_{\lambda}(I)$.

The main result in [11] for the setting $C_{\lambda}(\mathbb{S}^{1})$ is even
stronger than the preceding two statements.

######  Theorem 2.5.4.

The s-limit shadowing property is generic in $C_{\lambda}(\mathbb{S}^{1})$.

######  Corollary 2.5.5.

The limit shadowing, periodic shadowing and shadowing property are generic in
$C_{\lambda}(\mathbb{S}^{1})$.

Actually, Theorem 2.5.4 is somewhat surprising since
$C_{\lambda}(\mathbb{S}^{1})$ is the first environment in which s-limit
shadowing was proven to be generic. Up to now, only denseness of s-limit
shadowing was established in the setting of compact topological manifolds
[41]. Theorem 2.5.4 also holds in the setting of $C(\mathbb{S}^{1})$. However,
the methods used in [11] do not work in the setting of $C_{\lambda}(I)$. This
motivates the following.

######  Question A.

Is s-limit shadowing generic in $C_{\lambda}(I)$?

Possible positive answer to the above question will require some new
techniques than the ones used in [11]. On the other hand, a standard technique
to disprove that a condition is generic is to find an open set without the
required property. Such approach is again impossible, because of Proposition
2.5.3. In the view of Theorem 2.5.4 and the result from [31] it is also
natural to ask the following question.

######  Question B.

Is s-limit shadowing generic also for the volume preserving homeomorphisms on
manifolds of dimension greater than $1$?

In the context of volume preserving homeomorphisms on manifolds of dimension
at least two (with or without boundary), the genericity of shadowing was
recently proven by Guihéneuf and Lefeuvre [31].

###  2.6. Knot points

We define the upper, lower, left and right Dini derivatives of $f$ at $x$:

$\displaystyle D^{+}f(x)$ $\displaystyle:=\limsup_{\scriptstyle t\to
x^{+}}{\frac{f(t)-f(x)}{t-x}}\qquad D_{+}f(x):=\liminf_{\scriptstyle t\to
x^{+}}{\frac{f(t)-f(x)}{t-x}}$ $\displaystyle D^{-}f(x)$
$\displaystyle:=\limsup_{\scriptstyle t\to x^{-}}{\frac{f(t)-f(x)}{t-x}}\qquad
D_{-}f(x):=\liminf_{\scriptstyle t\to x^{-}}{\frac{f(t)-f(x)}{t-x}}.$

We call a point $x\in M$ a knot point of function $f\in C(M)$ if suprema and
infima of the right and left derivatives at point $x$ satisfy
$D^{+}f(x)=D^{-}f(x)=\infty$ and $D_{+}f(x)=D_{-}f(x)=-\infty$. The following
theorem states a consequence of a more general result proved in [10] for the
interval, the circle case can be treated analogously.

######  Theorem 2.6.1.

The $C_{\lambda}(M)$-generic function has a knot point at $\lambda$-almost
every point.

The next result generalizes a classical result of Saks [51] saying that the
set of Besicovitch functions is a meager set in $C(I)$. Its circle version
follows from the fact that a monotonicity result can be applied separately on
arcs partitioning the circle.

A Besicovitch function $f\in C(M)$ is a map such that for every $x\in M$, no
unilateral finite or infinite derivative exists at $x$.

######  Corollary 2.6.2.

The set of Besicovitch functions is a meager set in $C_{\lambda}(M)$.

###### Proof.

We use the following well known result (see [52, Theorem 7.3]): if for an arc
$A\subset M$ and $f\colon~{}A\to M$ we have $D^{+}f(x)\geq 0$ for a.e. $x\in
A$ and $D^{+}f(x)>-\infty$ for every $x\in A$, then $f$ is non-decreasing.

By Theorem 2.6.1 there is a residual set $K\subset C_{\lambda}(M)$ such that
each element of $K$ has a knot point at $\lambda$ almost every point of $M$.
Fix $f\in K$ and an arc $A\subset M$; we have $D^{+}f(x)=+\infty\geq 0$ a.e.
on $A$ hence $f$ can not be non-decreasing. Applying the above result, we
conclude that $D^{+}(x_{0})=-\infty$ for at least one point $x_{0}\in A$; in
particular $f$ is not a Besicovitch function. ∎

A Morse function $f\in C(M)$ satisfies

$\max\\{|D^{+}f(x)|,|D_{+}f(x)|\\}=\max\\{|D^{-}f(x)|,|D_{-}f(x)|\\}=\infty,~{}x\in
M;$

in the interval case, if $x$ is an endpoint of $I$, the only max is taken over
the derivatives from inside $I$. There exists a function $f\in C(M)$ which is
Besicovitch and Morse. The following question remains open.

######  Question C.

Does there exist a Besicovitch-Morse function in $C_{\lambda}(M)$?

###  2.7. Crookedness

We begin by defining the crookedness property which is one of the central
properties in Continuum Theory. We explain its importance later.

######  Definition 2.7.1.

Let $f\in C(I)$, $a,b\in I$ and let $\delta>0$. We say that $f$ is
$\delta$-crooked between $a$ and $b$ if for any two points $c,d\in I$ so that
$f(c)=a$ and $f(d)=b$, there is a point $c^{\prime}$ between $c$ and $d$ and
there is a point $d^{\prime}$ between $c^{\prime}$ and $d$ so that
$|b-f(c^{\prime})|<\delta$ and $|a-f(d^{\prime})|<\delta$. We will say that
$f$ is $\delta$-crooked if it is $\delta$-crooked between any pair of points.

In [23, Theorem 1] and [24] the authors proved the following generic property
of maps from $C_{\lambda}(M)$, which might be the most surprising of the
generic properties proven yet:

######  Theorem 2.7.2.

There is a dense $G_{\delta}$ set $\mathcal{T}\subset C_{\lambda}(M)$ such
that if $f\in\mathcal{T}$ then for every $\delta>0$ there exists a positive
integer $n$ so that $f^{n}$ is $(f,\delta)$-crooked.

The $\delta$-crookedness condition is a topological condition that imposes
strong requirements on values of the map. Piecewise smooth maps do not verify
the crookedness condition, thus Theorem 2.7.2 cannot hold for any open
collection of maps in $C_{\lambda}(M)$.

The pseudo-arc is a very curious object arising from Continuum Theory (see the
survey of Lewis [39] and the introduction of [18] for the overview of results
involving the pseudo-arc), which was first discovered by Knaster over a
century ago. On one hand side, its complicated structure is reflected by the
fact that it is hereditarily indecomposable, i.e., there are no proper
subcontinua $A,B\subset H$ such that $A\cup B=H$ for every proper subcontinuum
$H$ of the pseudo-arc $P$. On the other hand, the pseudo-arc is homogeneous,
i.e., for every two points $x,y\in P$ there exists a homeomorphism $h:P\to P$
such that $h(x)=y$. Homogeneity is a property possessed by the spaces with
locally identical structure. Non-trivial examples of homogeneous spaces are
the Cantor set, solenoids and manifolds without boundaries, for instance the
circle.

Let $\\{Z_{i}\\}_{i\geq 0}$ be a collection of compact metric spaces. For a
collection of continuous maps $f_{i}:Z_{i+1}\to Z_{i}$ we define

$\underleftarrow{\lim}(Z_{i},f_{i}):=\\{\hat{z}:=\big{(}z_{0},z_{1},\ldots\big{)}\in
Z_{0}\times Z_{1},\ldots\big{|}z_{i}\in Z_{i},z_{i}=f_{i}(z_{i+1}),\forall
i\geq 0\\}.$

We equip inverse limit $\underleftarrow{\lim}(Z_{i},f_{i})$ with the subspace
metric which is induced from the _product metric_ in $Z_{0}\times
Z_{1}\times\ldots$, where $f_{i}$ are called the bonding maps.

######  Corollary 2.7.3.

The inverse limit with any $C_{\lambda}(I)$-generic map as a single bonding
map is the pseudo-arc.

This corollary is a direct consequence of Theorem 2.7.2 and a result of Minc
and Transue [44, Proposition 4] connecting crookedness with pseudo-arc as
inverse limit.

###  2.8. Entropy

The property that the topological entropy of generic maps on the Lebesgue
measure-preserving maps is $\infty$ can be deduced from the methods of the
article of Yano [56]. Moreover, another way to see it from general theory is
to combine Theorem 2.7.3 with results of [45]. The connection between [56] and
$C_{\lambda}(I)$ was explicitly done in [17, Proposition 26]:

######  Theorem 2.8.1.

The generic value of topological entropy in $C_{\lambda}(I)$ is $\infty$.

The proof from [17] easily extends to $\mathbb{S}^{1}$, by replacing the fixed
point in the proof with a periodic point. The generic value of topological
entropy for the volume preserving continuous maps seems not to have been
studied for other manifolds. On the other hand Guihéneuf [29, Théorème 3.17]
proved the analogous result holds for generic homeomorphisms of compact,
connected manifold of dimension at least 2 preserving a “good” measure in the
sense of Oxtoby and Ulam [29]. The analogous result in the setting of
homeomorphisms on manifolds of dimension greater than $1$ was proven by Yano
[56]. Recently, Yano’s result was strengthened to show that the generic
homeomorphism or continuous map in most settings has an ergodic measure of
infinite entropy [22]. It would be interesting to know if this result holds
also in $C_{\lambda}(M)$.

On the other hand the question about generic value of metric entropy is
unclear. Let $PAM_{\lambda}(M)$ denote the set of piecewise affine Markov maps
that preserve Lebesgue measure. In [17, Proposition 24] the following theorem
is proven for $I$, the proof is analogous in the circle case:

######  Theorem 2.8.2.

For every $c\in(0,\infty)$ the set $PAM_{\lambda}(M)_{entr=c}$ is dense in
$C_{\lambda}(M)$.

Furthermore, the following theorem is proven in [17, Proposition 25] for
$C_{\lambda}(I)$ and can be analogously done with the help of Lemma 2.1.1 for
the circle case:

######  Theorem 2.8.3.

The set of maps from $C_{\lambda}(M)$ having metric entropy $\infty$ is dense
in $C_{\lambda}(M)$.

######  Question D.

Does there exist a generic value of metric entropy for maps from
$C_{\lambda}(M)$? If such value indeed exists, what is it? If there is no such
value, are all non-zero values attained by the metric entropy on every generic
set?

##  3\. Smoothness versus nowhere differentiability

###  3.1. Connections between topological and measure-theoretic properties

In [13] an effort has been made to understand natural conditions when
topological dynamical properties imply the corresponding measure-theoretic
properties in $C_{\lambda}(M)$ (and vice versa). The assumptions in the next
two theorems come directly from the articles of Li and Yorke [40] and Bowen
[19] respectively. The first result was proven in [13, Theorem 3].

######  Theorem 3.1.1.

Let $f\in C_{\lambda}(M)$ be a piecewise $C^{2}$ map with a slope strictly
greater than $1$. Then $f$ is transitive if and only if $(f,M,\lambda)$ is
ergodic.

It is well known that measure-theoretic exactness implies measure-theoretic
strong mixing, and since $\lambda$ is positive on open sets this furthermore
implies topological mixing. The first three points of the next result were
proven in [13, Theorem 2], which shows that these are equivalent in a smooth
enough one dimensional settings. They are also an important ingredient of the
proof of Theorem 2.2.3.

######  Theorem 3.1.2.

Let $f\in C_{\lambda}(M)$ be a piecewise $C^{2}$ map. Then the following
conditions are equivalent:

  1. (1)

$f$ is topologically mixing map,

  2. (2)

$f$ is strongly mixing,

  3. (3)

$(f,M,\lambda)$ is measure-theoretically exact,

  4. (4)

$f$ is leo.

To see that (4) is equivalent to (1)–(3) note that if $f\in C_{\lambda}(M)$
and is piecewise $C^{2}$ then it can have only a finite number of turning
points (which in fact, must be endpoints of pieces on which the map is
$C^{2}$). But by [34], a mixing interval map $f$ which is not leo has
infinitely many turning points.

Theorems 3.1.1 and 3.1.2 are in strong contrast to Theorem 2.4.6, which
displays a difference between piecewise smooth and non-differentiable
settings.

###  3.2. Nowhere differentiability, knot points and topological entropy

In [15] Bobok and Soukenka studied continuous piecewise affine interval maps
with countably many pieces of monotonicity that preserve the Lebesgue measure.
By taking limits of such maps they proved the following theorem:

######  Theorem 3.2.1.

There exists a map $g\in C_{\lambda}(I)$ such that:

  1. (1)

$g$ is nowhere monotone,

  2. (2)

knot points of g are dense in $I$ and for a dense $G_{\delta}$ set $Z$ of
$z$’s, the set $g^{-1}({z})$ is infinite;

  3. (3)

topological entropy $h_{\mathrm{top}}(g)\leq\mathrm{log}(2)+\varepsilon$.

Furthermore, the following two (yet unanswered) questions arose from their
study:

######  Question E.

Does every continuous nowhere differentiable interval map from
$C_{\lambda}(I)$ have infinite topological entropy?

The next question relates topological entropy with knot points (see Subsection
2.6).

######  Question F.

Does every map from $C_{\lambda}(I)$ with a knot point $\lambda$-a.e. have
infinite topological entropy?

Bobok and Soukenka continued their study in [16] where they studied a special
conjugacy class $\mathcal{F}$ of continuous piecewise monotone interval maps
with countably many laps (including Lebesgue measure-preserving maps), which
are locally eventually onto and all have topological entropy $\log(9)$. They
show that there exist maps from $\mathcal{F}$ with knot points in its fixed
point $1/2$.

##  4\. Other topologies

The aspects we discuss above are also interesting with other topologies on the
spaces of Lebesgue measure-preserving maps on one-dimensional compact
manifolds as well as higher dimensional analogues.

de Faria et. al. showed that topological entropy is infinite for
homeomorphisms in two different settings [27, 28]. As above, let $\mathbb{M}$
be compact $d$-dimensional manifold and $\mathcal{H}^{1}(\mathbb{M})$ be the
space of homeomorphisms which are bi-Lipschitz in all local charts. Let
$\mathcal{H}^{1}_{\alpha}$ denote the closure of $\mathcal{H}^{1}$ with
respect to the $\alpha$-Hölder-Whitney topology. Their first result is that
topological entropy is generically infinite in $\mathcal{H}^{1}_{\alpha}$
whenever $d\geq 2$ and $0<\alpha<1$.

For $1\leq p,p^{*}<\infty$ let $S^{p,p^{*}}(\mathbb{M})$ denote the space of
homeomorphisms on $\mathbb{M}$ which in all local charts are of Sobolev class
$W^{1,p}$ and whose inverse is of Sobolev class $W^{1,p^{*}}$ together with
the $(p,p^{*})$-Sobolev-Whitney topology. Their second result is that
topological entropy is generically infinite in $S^{p,p^{*}}(\mathbb{M})$ when
$d\geq 2$ and $d-1<p,p^{*}<\infty$.

In [35] Hazard constructed interesting examples of noninvertible maps with
infinite topological entropy in these topologies, however he did not study
generic behavior.

These results are the first dynamical genericity results for intermediate
smoothness. Generic values of topological entropy in these topologies has not
yet been studied in the volume preserving case. In fact no other dynamical
properties have been studied and it would be interesting to understand which
of the results of this survey hold in analogous topologies for continuous
Lebesgue measure preserving maps as well as simply for the continuous maps.

## Appendix

Let $\textrm{Rec}(f)$ denote the set of recurrent points of a map $f\in C(M)$.
It is proved in [36] that for graph maps (in particular for $\mathbb{S}^{1}$)
$\overline{\mathrm{Rec}(f)}=\overline{\mathrm{Per}(f)}\cup\mathrm{Rec}(f)$.
The following lemma confirms intuition that recurrence without dense
periodicity is possible only for irrational rotations of $\mathbb{S}^{1}$. The
following lemma is the crucial step in the proof of Remark 1.1 in the case of
$\mathbb{S}^{1}$.

######  Lemma A.1.

Let $f\in C(\mathbb{S}^{1})$. If $\overline{\mathrm{Rec}(f)}=\mathbb{S}^{1}$
and $\mathrm{Per}(f)\neq\emptyset$, then
$\overline{\mathrm{Per}(f)}=\mathbb{S}^{1}$.

###### Proof.

Assume towards contradiction that there is a nonempty open set
$U:=\mathbb{S}^{1}\setminus\overline{\mathrm{Per}(f)}$. Take a point $x\in
U\cap\mathrm{Rec}(f)$ and its maximal omega limit set $W\supset\omega(x)$.
Then by [7] (cf. [49]) $W$ is one of the following four types: a periodic
orbit, a basic set, a solenoidal set or a circumferential set. We claim that
none of these can occur, and so we have a contradiction, thus
$\overline{\mathrm{Per}(f)}=\mathbb{S}^{1}$.

Recall that $x\in\omega(x)$, so $W\cap U\neq\emptyset$, thus $W$ can not be a
periodic orbit. Since orbit of $x$ is infinite, there are nonnegative integers
$k<n<m$ such that $f^{k}(x),f^{n}(x),f^{m}(x)\in U$. If $W$ is a solenoidal
set, then there is a periodic interval $J$ such that each of these three
points belongs to a different iterate of $J$. In particular, there is a non-
negative integer $s$ such that $f^{s}(J)\subset U$. But since $J$ is a
periodic interval, $f^{s}(J)\cap\mathrm{Per}(f)\neq\emptyset$ which is again
impossible. If $W$ is a circumferential set, then there is a a connected set
$K\supset W$ and a monotone factor map $\phi\colon K\to\mathbb{S}^{1}$ semi-
conjugating $f|_{K}$ with an irrational rotation and such that each non-
singleton fiber of $\phi$ is a wandering interval. In particular, if $\phi$ is
not one-to-one then $\mathbb{S}^{1}\neq\overline{\textrm{Rec}(f)}$ which is a
contradiction. But if $\phi$ is one-to-one then $K$ is homeomorphic to
$\mathbb{S}^{1}$ which is again impossible, since $W=K$ is a subset of
$\mathbb{S}^{1}\setminus\mathrm{Per}(f)\neq\mathbb{S}^{1}$ but interval is not
homeomorphic to a circle. The last case is that $W$ is a basic set. By
repeating the argument from the circumferential set case, we can find a map
$\phi$ which conjugates $f^{n}|_{W}$ with a mixing interval map, for some $n$.
But a consequence of this conjugacy is that periodic points are dense in $W$,
in particular there is a periodic point arbitrarily close in $x$, so also in
$U$, which is again a contradiction. ∎

## Acknowledgements

The authors thank the European Regional Development Fund, project No. CZ
02.1.01/0.0/0.0/16_019/0000778 for financing their one week research meeting
in Prague where this survey article was finalized.

## References

  * [1] E. Akin, M. Hurley, J.A. Kennedy, Dynamics of topologically generic homeomorphisms, Mem. Amer. Math. Soc., 2003, 164, no. 783.
  * [2] E. Akin Stretching the Oxtoby-Ulam Theorem Colloquium Math. 84/85 (2000), 83–94.
  * [3] Ll. Alsedà, M.A. del Río, J.A. Rodríguez, Transitivity and dense periodicity for graph maps. J. Difference Equ. Appl. 9 (2003), no. 6, 577–598
  * [4] S. Alpern, V.S. Prasad, Typical dynamics of volume preserving homeomorphisms, Cambridge Tracts in Mathematics, 139, Cambridge University Press, Cambridge, 2000, 216 pp.
  * [5] S. Bezuglyi, J. Kwiatkowski, K. Medynets, Approximation in ergodic theory, Borel, and Cantor dynamics, Algebraic and topological dynamics, Contemp. Math., 385, Amer. Math. Soc., Providence, RI (2005), 39–64.
  * [6] L.S. Block, E.M. Coven, Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval, Trans. Amer. Math. Soc. 300 (1987) 297–306.
  * [7] A.M. Blokh, Dynamical systems on one-dimensional branched manifolds. I. (Russian) ; translated from Teor. Funktsii Funktsional. Anal. i Prilozhen. No. 46 (1986), 8–18 J. Soviet Math. 48 (1990), no. 5, 500–508
  * [8] A.M. Blokh, Spectral decomposition, periods of cycles and a conjecture of M. Misiurewicz for graph maps. Ergodic theory and related topics, III (Güstrow, 1990), 24–31, Lecture Notes in Math., 1514, Springer, Berlin, 1992
  * [9] A.M. Blokh, The “spectral” decomposition for one-dimensional maps. Dynamics reported, 1–59, Dynam. Report. Expositions Dynam. Systems (N.S.), 4, Springer, Berlin, 1995.
  * [10] J. Bobok, On non-differentiable measure-preserving functions, Real Analysis Exchange 16(1) (1991), 119–129.
  * [11] J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy, S-limit shadowing is generic for continuous Lebesgue measure-preserving circle maps, Erg. Theory Dyn. Sys., 43 (1), (2023) 78–98.
  * [12] J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy, Periodic points and shadowing property of Lebesgue measure-preserving interval maps, Nonlinearity 35 (2022) 2534–2557.
  * [13] J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy, Are generic dynamical properties stable under composition with rotations?, arXiv:2207.07186, July 2022.
  * [14] J. Bobok, J. Činč, P. Oprocha, S. Troubetzkoy, Generic maps with a dense set of periodic points, in preparation 2022.
  * [15] J. Bobok, M. Soukenka, Irreducibilty, infinite level sets and small entropy Real Analysis Exchange 36(2), 2010/2011, 449–462.
  * [16] J. Bobok, M. Soukenka, On piecewise affine interval maps with countably many laps. Disc. Cont. Dyn. Syst. 31(3), 2011, 753–762.
  * [17] J. Bobok, S. Troubetzkoy, Typical properties of interval maps preserving the Lebesgue measure, Nonlinearity, 33 (2020), 6461–6501.
  * [18] J. Boroński, J. Činč, P. Oprocha, Beyond $0$ and $\infty$: The solution to the Barge entropy conjecture, arXiv:2105.11133, May 2021.
  * [19] R. Bowen, Bernoulli maps of the interval, Isr. J. Math. 28 (1977), 161–168.
  * [20] J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc. 349 (1997), 2737–2754.
  * [21] M. Carvalho, F.B. Rodrigues and P. Varandas, Generic homeomorphisms have full metric mean dimension, Ergod. Theory Dyn. Syst. 42(1) (2022), 40–64.
  * [22] E. Catsigeras, S. Troubetzkoy, Ergodic measures with infinite entropy, arXiv:1901.07221, 2019.
  * [23] J. Činč, P. Oprocha, Parametrized family of pseudo-arc attractors: Physical measures and prime end rotations, Proc. London Math. Soc. (3) 125, (2022), 318–357.
  * [24] J. Činč, P. Oprocha, Parameterized family of pseudo-circle attractors, work in progress.
  * [25] E. M. Coven, G.A. Hedlund, $\overline{P}=\overline{R}$ for maps of the interval, Proc. Amer. Math. Soc. 79 (1980), 316–318.
  * [26] J.R. Choksi, V.S. Prasad, Approximation and Baire category theorems in ergodic theory, Measure theory and its applications (Sherbrooke, Que. 1982), 94–113, Lecture Notes in Math., 1033, Springer, Berlin, 1983.
  * [27] E. de Faria, Edson, P. Hazard, C. Tresser, Infinite entropy is generic in Hölder and Sobolev spaces. C.R. Math. Acad. Sci. Paris 355 (2017), no. 11, 1185–1189.
  * [28] E. de Faria, P. Hazard, C. Tresser, Genericity of infinite entropy for maps with low regularity, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (2021), no. 2, 601–664.
  * [29] P.-A. Guihéneuf, Propriétés dynamiques génériques des homéomorphismes conservatifs Ensaios Matematicos [Mathematical Surveys], 22, Sociedade Brasileira de Matemática, Rio de Janeiro, 115 pp, 2012.
  * [30] P.-A. Guihéneuf, Ensemble des points périodiques d’un homéomorphisme conservatif générique https://webusers.imj-prg.fr/~pierre-antoine.guiheneuf/Fichiers/PerGeneHom.pdf
  * [31] P.-A. Guihéneuf, T. Lefeuvre, On the genericity of the shadowing property for conservative homeomorphisms, Proc. Am. Math. Soc. 146 (2018), 4225–4237.
  * [32] P.R. Halmos, In general a measure-preserving transformation is mixing, Ann. of Math. 45(2), (1944), 786–792.
  * [33] P.R. Halmos, Approximation theories for measure-preserving transformations, Trans. Amer. Math. Soc. 55 (1944), 1–18.
  * [34] G. Harańczyk, D. Kwietniak, P. Oprocha, Topological structure and entropy of mixing graph maps, Ergodic Theory Dynam. Systems 34 (2014), no. 5, 1587–1614.
  * [35] P. Hazard, Maps in dimension one with infinite entropy. Ark. Mat. 58 (2020), no. 1, 95–119.
  * [36] J.-H. Mai and S. Shao, $\overline{R}=R\cup\overline{P}$ for graph maps. J. Math. Anal. Appl. 350 (2009), no. 1, 9–11.
  * [37] A. Katok, A. Stepin, Approximations in ergodic theory, Uspehi Mat. Nauk 22 (1967), 81–106.
  * [38] P. Koscielniak, M. Mazur, P. Oprocha, L. Kubica, Shadowing is Generic on Various One-Dimensional Continua with a Special Geometric Structure, J. Geom. Anal. 30(2) (2020), 1836–1864.
  * [39] W. Lewis, The pseudo-arc. Continuum theory and dynamical systems (Arcata, CA, 1989), 103–123, Contemp. Math., 117, Amer. Math. Soc., Providence, RI, 1991.
  * [40] T.Y. Li, J.A. Yorke, Ergodic transformation from an interval into itself, Trans. Amer. Math. Soc. 235(1978), 183–192.
  * [41] M. Mazur, P. Oprocha, S-limit shadowing is $\mathcal{C}^{0}$-dense, J. Math. Anal. Appl. 408(2) (2013), 465–475.
  * [42] I. Mizera, Generic properties of one-dimensional dynamical systems, Ergodic theory and related topics, III (Guüstrow, 1990), Lecture Notes in Math. 1514 (1992), Springer, Berlin, 163–173.
  * [43] J. Meddaugh, On genericity of shadowing in one-dimensional continua, Fund. Math. 255 (2021), 1–18.
  * [44] P. Minc, W. R. R. Transue, _A transitive map on $I$ whose inverse limit is the pseudo-arc_, Proc. Amer. Math. Soc. 111, no. 4 (1991), 1165–1170.
  * [45] C. Mouron, Entropy of shift maps of the pseudo-arc. Topology Appl. 159 (2012), no. 1, 34–39.
  * [46] J. Oxtoby, S. Ulam, Measure-preserving homeomorphisms and metric transitivity, Ann. Math. 42 (1941), 874–920.
  * [47] K. Petersen, Measure Preserving Systems, Encyclopedia of Complexity and Systems Science, 5450–5466.
  * [48] S.Yu. Pilyugin, O.B. Plamenevskaya, Shadowing is generic,Topology Appl. 97(3) (1999), 253–266.
  * [49] S. Ruette, L. Snoha, For graph maps, one scrambled pair implies Li-Yorke chaos. Proc. Amer. Math. Soc. 142 (2014), no. 6, 2087–2100.
  * [50] V. Rohlin, A “general” measure-preserving transformation is not mixing, Doklady Akad. Nauk SSSR (N.S.) 60 (1948), 349–351.
  * [51] S. Saks, On the functions of Besicovitch in the space of continuous functions, Fundamenta Mathematicae 19 (1932), 211–219.
  * [52] S. Saks, On the functions of Besicovitch in the space of continuous functions, Fundamenta Mathematicae 19 (1932), 211–219.
  * [53] J. Schmeling, R. Winkler, Typical dimension of the graph of certain functions, Monatsh. Math. 119 (1995), 303-–320.
  * [54] S. Troubetzkoy, Approximation and billiards, Dynamical systems and Diophantine approximation, 173–185, Sémin. Congr., 19, Soc. Math. France, Paris, 2009.
  * [55] P. Walters, An Introduction to Ergodic Theory (New York: Springer), 1982.
  * [56] K. Yano, A remark on the topological entropy of homeomorphisms, Inv. Math. 59 (1980), 215–220.