# A geometric framework for asymptoticity and expansivity in topological
dynamics

Sebastián Donoso  Departamento de Ingeniería Matemática and Centro de
Modelamiento Matemático, Universidad de Chile and IRL-CNRS 2807, Beauchef 851,
Santiago, Chile<EMAIL_ADDRESS>,  Alejandro Maass  Departamento de
Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de
Chile and IRL-CNRS 2807, Beauchef 851, Santiago, Chile<EMAIL_ADDRESS>and  Samuel Petite  Laboratoire Amiénois de Mathématiques Fondamentales et
Appliquées, CNRS-UMR 7352, Université de Picardie Jules Verne, 33 rue Saint
Leu, 80039 Amiens cedex 1, France<EMAIL_ADDRESS>
###### Abstract.

In this paper we develop a geometric framework to address asymptoticity and
nonexpansivity in topological dynamics when the acting group is second
countable and locally compact. As an application, we show extensions of
Schwartzman’s theorem in this context. Also, we get new results when the
acting groups is ${\mathbb{Z}}^{d}$: any half-space of ${\mathbb{R}}^{d}$
contains a vector defining a (oriented) nonexpansive direction in the sense of
Boyle and Lind. Finally, we deduce rigidity properties of distal Cantor
systems.

###### Key words and phrases:

asymptoticity, expansivity, horoballs

###### 2020 Mathematics Subject Classification:

Primary: 37B05; Secondary: 54H15, 20F65, 37B10, 37B20

The first and second authors were partially funded by Centro de Modelamiento
Matemático (CMM), ACE210010 and FB210005, BASAL funds for centers of
excellence from ANID-Chile. The third author was partially funded by the ANR
Project IZES ANR-22-CE40-0011. All authors are part of the ECOS-ANID grant
C21E04 (ECOS210033). The first author was partially funded by
ANID/Fondecyt/1200897

##  1\. Introduction

Let $T\colon X\to X$ be a self-homeomorphism of an infinite compact metric
space $X$. S. Schwartzman proved in [28] the following celebrated result:
given $\epsilon>0$, there exist different points $x,y\in X$ such that for all
$n\geq 0$ one has

${\rm dist}(T^{n}x,T^{n}y)\leq\epsilon,$

where ${\rm dist}$ is the distance in $X$. That is, there exist positive
$\epsilon$-asymptotic pairs. Considering $T^{-1}$ in place of $T$, the same
result implies that there exist negative $\epsilon$-asymptotic pairs 111This
means that there are two different points that are $\epsilon$-close for all
$n\leq 0$.. Another way to state the same result is by saying that in an
infinite compact metric space it is impossible to construct positively
expansive homeomorphisms. This result has been recovered independently and in
different flavours by several authors (see for instance [1, 5, 7, 14, 18, 19,
23, 27] and the survey [9]).

Motivated by the study of subactions of higher rank abelian actions, M. Boyle
and D. Lind obtained in [7] a major result in the spirit of Schwarztman’s
theorem. They introduced the notion of nonexpansivity along a subspace of
${\mathbb{R}}^{d}$, by considering only the action of elements of
${\mathbb{Z}}^{d}$ that lie within a given bounded distance of the subspace.
They showed (see also [12]) that for an infinite compact metric space $X$,
there exist a half-space $H\subseteq{\mathbb{R}}^{d}$ and different points
$x,y$ in $X$ that stay arbitrarily close along their
$H\cap{\mathbb{Z}}^{d}$-orbit (see Section 5). Such half-space is called
nonexpansive since the action of its border is nonexpansive.

Let us point out that unlike Schwartzman’s result, where the two half-spaces
of ${\mathbb{Z}}$ are nonexpansive, here only one nonexpansive half-space is
guaranteed to exist. Actually, not any half-space is nonexpansive for an
arbitrary ${\mathbb{Z}}^{2}$ system. Explicit examples like Ledrappier’s
${\mathbb{Z}}^{2}$ subshift illustrate that the set of such half-spaces can
even be finite. A motivation for studying the set of nonexpansive half-spaces
is that many dynamical properties of the subactions vary nicely along the
connected components of the set of expansive directions and a bifurcation
phenomenon may occur when passing from one component to another.

To complete their result, Boyle and Lind, and later M. Hochman, realized any
nonempty closed set of half-spaces as the set of nonexpansive ones of some
${\mathbb{Z}}^{2}$-action. In particular, Hochman succeeded in realizing the
challenging case where this set consists of a single half-space whose border
line has an irrational slope [17].

The notions of nonexpansive half-spaces and asymptotic points have turned out
to be the key notions to tackle various problems. For instance, in dynamics,
they appear as fundamental objects for studying expansive maps (e.g., [21]),
their centralizer and cellular automata (e.g., [25, 8]), or to study
topological joinings (e.g., [19]). For ${\mathbb{Z}}^{d}$-actions, they are
main notions to explore directional invariants like directional entropy (e.g.,
[26]). They also appear in combinatorics when addressing the long standing
Nivat conjecture (e.g., [11]). Despite the breadth of possible applications,
no similar concept has been proposed for general group actions.

The definition and the proofs in [7] seem to rely heavily on the linear
structure of ${\mathbb{R}}^{d}$, so extensions of these results to general
group actions were open for a long time. In this work, we develop a framework
using basic elements of geometric group theory that allows us to give a
definition of nonexpansiveness for actions of countable groups and more
generally for second countable and locally compact topological groups. The
notion of half-space is replaced by that of horoball (see Section 2 for the
precise definition). Horoballs are sublevel sets of horofunctions, which are
the analogous to linear forms on general metric spaces. They depend on the
choice of a proper right-invariant distance on the acting group. We warn the
reader that, depending on the group and the metric, horoballs can contain
arbitrary large balls, like in the case of finitely generated groups, or can
even be empty, like in the case of infinite sum of finite groups (see 2.1 and
the discussion around it). Anyway, using this framework, we show the following
extension of Boyle and Lind’s results, which is meaningful for a large class
of groups including the finitely generated ones.

######  Theorem 1.1.

Let $(X,T,G)$ be an infinite topological dynamical system and $G$ be an
infinite group with a proper right-invariant distance. Then, for all
$\epsilon>0$ there exist different points $x,y\in X$ and a horoball $H$ such
that ${\rm dist}(T_{g}x,T_{g}y)\leq\epsilon$ for all $g\in H$.

Actually, 1.1 is derived from the following more general theorem, which we
believe is of independent interest and might be useful for other problems in
topological dynamics.

######  Theorem 1.2 (Robinson Crusoe theorem).

Let $(X,T,G)$ be a topological dynamical system and $G$ be an infinite group
with a proper right-invariant distance. Let $O\subsetneq X$ be an open, not
closed and $G$-invariant ($T_{g}(O)=O,\forall g\in G$) subset of $X$. Then,
for any neighborhood $U$ of the boundary of $O$ there exists a horoball $H$ in
$G$ such that

$O\cap\bigcap_{g\in H}T_{g}^{-1}(U)\neq\emptyset.$

We name this theorem Robinson Crusoe because of the following interpretation.
Assume that Robinson Crusoe is isolated on an island and his movement is
deterministic, i.e., given by a ${\mathbb{R}}$-flow. The island defines a
${\mathbb{R}}$-invariant open set $O$ in the phase space. The theorem claims
that there exists a trajectory of Robinson Crusoe that stays on the beach (the
border of the island) for all positive (or negative) times. This can be seen
as a containment result: $O$ is not repulsed by its border for positive or
negative times.

From 1.2 we deduce in 4.3 the existence of $\epsilon$-asymptotic pairs
relatively to a factor map, and from this 1.1 follows immediately. When
$G={\mathbb{Z}}^{d}$ is endowed with the Euclidean metric, we recover the
theorem of Boyle and Lind on the existence of nonexpansive half-spaces.

Pursuing extensions of Schwartzman’s result, we also provide a directed
version of Robinson Crusoe theorem where the set of horoballs can be
restricted to a smaller family, as long as the action satisfies a non-
repulsive condition along a semi-group (3.4). Although this technical
condition restricts the actions, the family of systems that satisfy it is
rather broad. We deduce in 4.4 a directed version of asymptotic pairs for
infinite groups admitting a proper right-invariant distance such that the
inversion map $g\mapsto g^{-1}$ is an isometry. As a consequence, for
$G={\mathbb{Z}}^{d}$ we obtain the following new restriction on the
nonexpansive half-spaces of a given action (see 5.6).

######  Theorem 1.3.

Let $(X,T,{\mathbb{Z}}^{d})$ be an infinite topological dynamical system.
Then, the intersection of all its nonexpansive open half-spaces is empty.

Whereas 1.1 states that there exists at least one nonexpansive half-space for
${\mathbb{Z}}^{d}$-actions, 1.3 establishes that the half-space cannot be
unique. In particular, when $d=1$, these recover Schwartzman’s result. Thus,
Theorems 1.1 and 1.3 constitute the first known common generalization of
Schwartzman’s and Boyle and Lind’s results.

Finally, we apply Robinson Crusoe theorem to Cantor dynamics to get
restrictions on distal systems and distal factors for a class of groups
containing finitely generated groups: they have to be equicontinuous (6.2).
This recovers results in [4] and in [23] for subshifts.

### Organization of the paper

In Section 2 we introduce the basic tools we need from geometric group theory.
Section 3 is devoted to proving the main results of the paper: the Robinson
Crusoe theorem (3.1) and the Directed Robinson Crusoe theorem (3.4). Then, in
Section 4 we derive applications of these theorems to the notions of
nonexpansiveness in topological dynamics. In Section 5 we focus on
${\mathbb{Z}}^{d}$-actions, and in particular we show how to obtain 1.3 (5.6).
In this section we also discuss the dependency of our main theorems on the
metric chosen in ${\mathbb{Z}}^{d}$ and show interesting phenomena. The last
section, Section 6, is devoted to applications in Cantor dynamics that are of
independent interest.

## Acknowledgement

We are grateful to Benjamin Weiss and Xiangdong Ye for pointing out an error
in an earlier version of this paper, namely that 6.3 does not hold for any
countable group.

##  2\. Preliminaries

###  2.1. Basics on groups geometry

In this section, we recall some basic facts on horoballs on groups. Given a
group $G$ we denote by $1_{G}$ its neutral element. In what follows we will
assume that the group $G$ is topological, second countable and locally
compact. Equivalently (see [29]), it admits a distance $d\colon G\times
G\to{\mathbb{R}}$ which is

  * •

right-invariant, i.e., $d(gf,hf)=d(g,h)$ for all $g,h,f\in G$;

  * •

and proper, i.e., every closed ball is compact.

We remark that even if we will work with a distance, all the results presented
in this paper could be extended to any continuous right-invariant and proper
semi-distance.

Any countable group satisfies these hypotheses for an integer valued distance.
For instance, when $G$ is a group generated by a finite set $\mathcal{S}$, a
proper right-invariant distance is given by the $\ell_{1}$ distance:
$d_{\ell_{1}}(g,h)=\|gh^{-1}\|_{1}=\inf\\{n\in{\mathbb{N}}:gh^{-1}=s_{1}\cdots
s_{n}\text{ for }s_{1},\ldots,s_{n}\in{\mathcal{S}}\cup{\mathcal{S}}^{-1}\\}$.
In the general case (e.g., when $G$ is not finitely generated), the classical
Higman-Neumann-Neumann Theorem ensures that $G$ is a subgroup of a finitely
generated group so that the former distance induces a proper right-invariant
distance on $G$. A simple and constructive way is the following. Consider a
sequence of finite subsets $(B_{n})_{n\geq 0}$ of $G$ such that

  * •

$B_{0}=\\{1_{G}\\}$;

  * •

each $B_{n}$ is symmetric, i.e., $B_{n}=B_{n}^{-1}$;

  * •

$B_{n}B_{m}\subseteq B_{n+m}$ for any $n,m\geq 0$;

  * •

$G=\bigcup_{n\geq 0}B_{n}$.

Then, $\tilde{d}(g,h)=\inf\\{n\in{\mathbb{N}}:gh^{-1}\in B_{n}\\}$ defines a
proper right-invariant distance and each $B_{n}$ is the open ball of radius
$n$ centered at the neutral element $1_{G}$ for this distance.

Generalizing the ideas of H. Busemann, M. Gromov defines a compactification of
the group $G$ with an embedding map $b$. Denoting by $C(G)$ the collection of
continuous real functions on $G$, this embedding is given by,

$\displaystyle b\colon G$ $\displaystyle\to$ $\displaystyle C(G)$
$\displaystyle g$ $\displaystyle\mapsto$ $\displaystyle b_{g}\colon x\mapsto
d(g,x)-d(g,1_{G}).$

The triangle inequality implies that all the maps $b_{g}$ are 1-Lipschitz.
Moreover, by construction, we have $b_{g}(1_{G})=0$. It follows from Arzelà-
Ascoli’s theorem and a diagonal argument that $b(G)$ is a relatively compact
set in $C(G)$ for the compact-open topology. It is straightforward to check
that the map $b$ is an injection.

The border of $G$, denoted by $\partial G$, is the set

$\partial G=\overline{b(G)}\setminus b(G),$

where the closure is considered for the compact-open topology (i.e., here it
corresponds to the uniform convergence on compact sets, which is first
countable). This set is not empty if $G$ is unbounded. Any function
$j\in\partial G$ is called an horofunction (or a Busemann cocycle). Moreover,
there exists a sequence $(g_{n})_{n\in{\mathbb{N}}}\subseteq G$ going to
infinity 222This means that for any $m$, $g_{n}\notin B_{m}(1_{G})$ for all
large enough $n$, i.e., $g_{n}$ eventually leaves any ball. such that
$\displaystyle j=\lim_{n\to\infty}b_{g_{n}}$. We call horoball any subset of
$G$ of the form

$H=\\{x\in G:\ j(x)<0\\}\textrm{ for some element }j\in\partial G.$

We denote by $\mathcal{H}$ the collection of all horoballs. Notice that each
horoball can be seen as the limit of sets of the form $\\{g\in
G:b_{g_{n}}(g)<0\\}$, i.e., open balls centered at $g_{n}$ with radius
$d(g_{n},1_{G})$.

$\times$$O$$\times$$g_{1}$$\times$$g_{2}$$\times$$g_{3}$ Figure 1. A
$\ell_{2}$ horoball (in gray) for the group ${\mathbb{Z}}^{2}$ obtained as a
limit of balls.

The notion of horoball depends on the distance $d$. For instance, for the
$\ell_{2}$ norm on ${\mathbb{Z}}^{d}$, a standard computation shows that any
horofunction which is the limit of functions $b_{g_{n}}$, with $g_{n}$ going
to infinity, is of the form $\langle v,\cdot\rangle$ for some vector
$v\in{\mathbb{R}}^{d}$ that is the accumulation point of the vectors
$g_{n}/\|g_{n}\|_{2}$. Conversely, any unit vector $v$ can be obtained in this
way. Hence, a horoball is an open half-space delimited by a hyperplane, as
illustrated in Fig. 1. In the sequel we will identify the $\ell_{2}$ horoball
$\\{x\in{\mathbb{Z}}^{d}:\langle x,v\rangle<0\\}$ with its outgoing normal
vector $v$. The set of $\ell_{2}$ horoballs is then naturally parameterized by
the unit sphere ${\mathbb{S}}^{d-1}\subseteq{\mathbb{R}}^{d}$.

For the $\ell_{1}$ norm on ${\mathbb{Z}}^{d}$ the set of horoballs looks
really different since it is countable. It can be checked by considering the
horoballs as limits of $\ell_{1}$ balls with centers going to infinity and by
analyzing the ultimate position of the origin on these $\ell_{1}$ balls. For
instance, in ${\mathbb{Z}}^{2}$, the horoballs are half-spaces delimited by
the diagonal or the anti-diagonal lines, and quarter spaces delimited by
integer translations of such half-spaces (see Fig. 2). A similar phenomenon
occurs for the $\ell_{\infty}$ norm. However, notice that any horoball for a
norm $\ell_{p}$, $p\in[1,+\infty]$, is included in a $\ell_{2}$ horoball.

$\times$$O$ Figure 2. A $\ell_{1}$ horoball (in gray) for the group
${\mathbb{Z}}^{2}$ obtained as limit of balls.

Unlike the euclidean case, the spaces of horoballs are all homeomorphic when
considering hyperbolic distances on a fixed hyperbolic group [16].

In the following, $B_{R}(g)=\\{h\in G:d(h,g)<R\\}$ denotes the open ball of
radius $R>0$ centered at $g\in G$ for the metric $d$ and its closure, which is
compact by assumption, is denoted by $\overline{B_{R}(g)}$. It is worth noting
that $B_{R}(g)=B_{R}(1_{G})g$, since the distance is right-invariant. In
addition, any ball at the origin is symmetric, that is,
$B_{R}(1_{G})^{-1}=B_{R}(1_{G})$.

Depending on the chosen distance on the group $G$, the horoball could be
large, i.e., contains arbitrary large balls, or could be even empty. For
instance, any horoball of ${\mathbb{Z}}^{d}$ or ${\mathbb{R}}^{d}$ for the
$\ell_{p}$ norms, $p\in[1,+\infty]$, are large. We provide here a criterion
that assures to have large horoballs. This is a strengthening of [4,
Proposition 1.5]. This applies particularly to horoballs associated with the
$\ell_{1}$ distance of a finitely generated group.

######  Proposition 2.1.

Let $G$ be an infinite group with a proper right-invariant distance. If a
horofunction $j$ satisfies $\inf_{g\in G}j(g)=-\infty$, then its associated
horoball contains arbitrarily large balls.

As a consequence, any horoball for the $\ell_{1}$ distance of a finitely
generated group contains arbitrarily large balls.

###### Proof.

Let $R>0$ and $(g_{n})_{n\in{\mathbb{N}}}\subseteq G$ be a sequence such that
$b_{g_{n}}$ converges to $j$. By hypothesis, we can take $g\in G$ such that
$j(g)<-4R$. Then, for any large enough $n\in{\mathbb{N}}$,
$b_{g_{n}}(g)\leq-2R$. Now, for any $h\in B_{R}(g)$, the triangular inequality
gives

$b_{g_{n}}(h)=d(h,g_{n})-d(g_{n},1_{G})\leq
d(h,g)+d(g_{n},g)-d(g_{n},1_{G})\leq-R,$

where the last inequality is true for any large enough $n$. This shows that
the ball $B_{R}(g)$ is contained in the horoball $\\{j<0\\}$.

To prove the second statement of the lemma, assume that $G$ is a group
generated by a finite set ${\mathcal{S}}$ and consider its $\ell_{1}$
distance. Observe that for any sequence $(g_{n})_{n\in{\mathbb{N}}}\subseteq
G$ leaving any compact, the element $g_{n}$ is given by
$s_{N_{n}}^{(n)}s_{N_{n}-1}^{(n)}\ldots s_{1}^{(n)}$ with
$s_{j}^{(n)}\in{\mathcal{S}}\cup{\mathcal{S}}^{-1}$, where $N_{n}$ is the
smallest such integer, i.e., $N_{n}=d_{\ell_{1}}(g_{n},1_{G})$. For any
integer $k>0$, up to taking a subsequence, the term
$s_{k}^{(n)}s_{k-1}^{(n)}\ldots s_{1}^{(n)}$ is constant in $n$. It is then
denoted $s_{k}\ldots s_{1}$. For any large enough $n$, the very definition of
the $\ell_{1}$ distance provides

$b_{g_{n}}(s_{k}\ldots s_{1})=d_{\ell_{1}}(g_{n},s_{k}\ldots
s_{1})-N_{n}=d_{\ell_{1}}(g_{n}s_{1}^{-1}\ldots
s_{k}^{-1},1_{G})-N_{n}\leq-k.$

Since $k$ is arbitrary, this shows that any horofunction $j$ that is an
accumulation point of $b_{g_{n}}$ satisfies $\inf_{g\in G}j(g)=-\infty$. ∎

On the contrary, some distances may define horofunctions with empty horoballs.
For instance, consider the free abelian group infinitely generated by
$(e_{i})_{i\in{\mathbb{N}}}$ (i.e.,
$\bigoplus_{i\in{\mathbb{N}}}{\mathbb{Z}}$), equipped with the norm
$\|\sum_{i\in{\mathbb{N}}}x_{i}e_{i}\|=\sum_{i\in{\mathbb{N}}}|x_{i}|i$ with
$x_{i}\in{\mathbb{Z}}$. Then, any horofunction that is an accumulation point
of the functions $b_{e_{i}}$ has a trivial horoball. More dramatically, for
the locally finite group
$\bigoplus_{i\in{\mathbb{N}}}{\mathbb{Z}}/2{\mathbb{Z}}$, the same remark
provides that any horoball is empty.

Throughout the proofs, the most interesting case will be when the group $G$
does not have empty horoballs. A compactness argument provides the following
finitary characterization of this case.

######  Lemma 2.2.

Let $G$ be a group with a proper right-invariant distance that does not admit
empty horoballs. Then, there exists an integer $N>0$ such that

$\displaystyle H\cap B_{N}(1_{G})\neq\emptyset,\quad\forall
H\in{\mathcal{H}}.$

###### Proof.

For the sake of a contradiction, assume that there is a sequence of
horofunctions $(j_{n})_{n\geq 1}$ such that each horoball $\\{x:j_{n}(x)<0\\}$
is disjoint from the ball $B_{n}(1_{G})$. Let $j\in\partial G$ be an
accumulation point of the sequence $(j_{n})_{n\geq 1}$. The horoball
$\\{x:j(x)<0\\}$ is then disjoint from any ball centered at $1_{G}$, hence is
empty. This contradicts the hypothesis. ∎

Our applications in Section 5 will mainly focus on locally compact, second
countable groups that are subgroups of a Banach space $E$ (hence are abelian).
In this case, it is known that any horofunction is bounded from below by a
linear functional $L$ of norm at most $1$ [13, Lemma 3.1]. Hence, any horoball
is included in a half-space of the form $L<0$. In this sense, the half-spaces
are the largest possible horoballs for $G$. For a general group, this
geometrical property is still valid but only locally: any horoball is locally
tangent to a ball, in the identity element. See 2.4 and its illustration in
Fig. 3.

Also remark that for a countable group $G$, the ball $B_{\epsilon}(1_{G})$ is
trivial, i.e., reduced to the neutral element $1_{G}$, for any small enough
$\epsilon$. So, the reader interested only in countable groups may simplify
the formulas in all the following statements.

######  Lemma 2.3.

Let $G$ be an infinite group with a proper right-invariant distance $d$. Then,
for any $M>0$ and $\epsilon>0$ there exists an integer $n_{0}\in{\mathbb{N}}$
such that for $g\in G$ with $d(g,1_{G})\geq n_{0}$ one can find a horoball
$H\in{\mathcal{H}}$ such that

$\left[\overline{H}\cap\overline{B_{M}(1_{G})}\right]g\subseteq
B_{\epsilon}(1_{G})B_{d(g,1_{G})}(1_{G}).$

2.3 is a particular case of 2.4 below, where we do not need to consider all
the horoballs but only those that are defined by taking the limit of the
points outside a specific subset of $G$. We introduce the following notions
and notations. For an unbounded set $G_{0}\subseteq G$, we denote by $\partial
G_{0}$ the set of horofunctions $\displaystyle
j=\lim_{n\to\infty}b_{g_{n}}\in\partial G$, where $(g_{n})_{n\in{\mathbb{N}}}$
is a sequence in $G_{0}$ going to infinity. Similarly, we denote by
${\mathcal{H}}_{\partial G_{0}}$ the set of horoballs $\\{x\in G:j(x)<0\\}$
for $j\in\partial G_{0}$. Note that 2.3 follows from 2.4 by taking $A=G$ and
$\bar{g}=1_{G}$.

######  Lemma 2.4.

Let $G$ be an infinite group with a proper right-invariant distance $d$ and
$A\subseteq G$ be an unbounded subset. Then, for any $M>0$ and $\epsilon>0$
there exists an integer $n_{0}\in{\mathbb{N}}$ such that for any $\bar{g}\in
G$ and $g\in A\bar{g}$ with $d(g,\bar{g})\geq n_{0}$ one can find a horoball
$H\in{\mathcal{H}}_{\partial A^{-1}}$ such that

$\left[\overline{H}\cap\overline{B_{M}(1_{G})}\right]g\subseteq
B_{\epsilon}(1_{G})B_{d(g,\bar{g})}(\bar{g}).$

$Hg$$M$$\times$$\bar{g}$$\epsilon$$\times$$g$ Figure 3. Illustration of 2.4
for the $\ell_{2}$ distance on $G={\mathbb{Z}}^{2}$.

###### Proof.

We first show that the result holds for $\bar{g}=1_{G}$. By contradiction,
suppose that there exist constants $M>0$ and $\epsilon>0$ and a sequence
$(g_{n})_{n\in{\mathbb{N}}}$ in $A$ going to infinity, such that for any
horoball $H\in{\mathcal{H}}_{\partial A^{-1}}$ we have that

(1)
$\displaystyle\left[\overline{H}\cap\overline{B_{M}(1_{G})}\right]\not\subseteq
B_{\epsilon}(1_{G})B_{d(g_{n},1_{G})}(g_{n}^{-1}).$

Up to taking a subsequence, we can assume that
$(b_{g_{n}^{-1}})_{n\in{\mathbb{N}}}$ converges to a horofunction
$j\in\partial A^{-1}$. By assumption, the horoball
$H=\\{j<0\\}\in{\mathcal{H}}_{\partial A^{-1}}$ satisfies (1). By a
compactness argument, there exists an element
$x\in\overline{H}\cap\overline{B_{M}(1_{G})}$ such that for infinitely many
$n\in{\mathbb{N}}$, $d(x,B_{d(g_{n},1_{G})}(g_{n}^{-1}))\geq\epsilon$. Take an
element $z\in H$ such that $d(x,z)<\epsilon/2$. For any large enough
$n\in{\mathbb{N}}$ we have that $b_{g_{n}^{-1}}(z)<0$ (i.e., $z\in
B_{d(g_{n},1_{G})}(g_{n}^{-1})$), but by the choice of $x$, for infinitely
many $n\in{\mathbb{N}}$ we have that
$d(z,B_{d(g_{n},1_{G})}(g_{n}^{-1}))\geq\epsilon/2$. This is a contradiction
and we therefore obtain the result for $\bar{g}=1_{G}$.

To conclude the general case for $\bar{g}\in G$ and $g\in A\bar{g}$, it is
enough to use the case $\bar{g}=1_{G}$ taking $g\bar{g}^{-1}$ in place of $g$.
∎

The next geometrical lemma is needed for the directed version of our main
result. Roughly speaking, it states that some translation of a “cone” $G_{0}$
has a larger intersection with a ball around the origin than that of the
untranslated cone $G_{0}$. See Fig. 4 for an illustration.

######  Lemma 2.5.

Let $G$ be an infinite group with a proper right-invariant distance $d$,
$G_{0}\subseteq G$ an unbounded subset and $\eta>0$. Then, for any $g\in G$
such that $j(g^{-1})<-\eta$ for any $j\in\partial G_{0}$, there exists an
integer $n_{1}\in{\mathbb{N}}$ such that for any real number $r\geq n_{1}$

$\left[G_{0}\cap B_{r+\eta}(1_{G})\right]g\subseteq B_{r}(1_{G}).$

###### Proof.

The proof is by contradiction. If the result does not hold, there exist $g\in
G$ with $j(g^{-1})<-\eta$ for all $j\in\partial G_{0}$ and sequences
$(g_{n})_{n\in{\mathbb{N}}}$ in $G_{0}$ and $(r_{n})_{n\in{\mathbb{N}}}$ in
${\mathbb{R}}$ going to infinity such that $d(g_{n},1_{G})<r_{n}+\eta$ and
$d(g_{n}g,1_{G})\geq r_{n}$ for all $n\in{\mathbb{N}}$. This means that
$b_{g_{n}}(g^{-1})\geq-\eta$. Since the elements $g_{n}$ go to infinity, any
accumulation point $j\in\partial G_{0}$ satisfies $j(g^{-1})\geq-\eta$. This
is a contradiction. ∎

$r$$\times$$0$$G_{0}$$\times$$g$$r+\eta$$G_{0}g$ Figure 4. Illustration of 2.5
for $G={\mathbb{Z}}^{2}$.

##  3\. Robinson Crusoe theorem

In this section we prove our main abstract theorems that will allow us in the
next sections to establish some extensions of Schwartzman’s theorem and other
applications. We recall that a topological dynamical system $(X,T,G)$ is given
by a continuous left-action $T\colon G\times X\to X$ of a group $G$ on a
compact metric space $X$ equipped with a distance ${\rm dist}$. This provides
a family of self-homeomorphisms $\\{T_{g}:g\in G\\}$ of $X$ such that
$T_{g}\circ T_{h}=T_{gh}$ for any $g,h\in G$ and the maps $T_{g}$ depend
continuously (for the uniform topology) on the element $g$. A subset
$Y\subseteq X$ is $G$-invariant (or simply invariant when there is no
ambiguity) if $T_{g}(Y)=Y$ for all $g\in G$.

The next theorem is 1.2 from the introduction.

######  Theorem 3.1 (Robinson Crusoe theorem).

Let $(X,T,G)$ be a topological dynamical system and $G$ be an infinite group
with a proper right-invariant distance. Let $O\subsetneq X$ be an open, not
closed and $G$-invariant ($T_{g}(O)=O,\forall g\in G$) subset of $X$. Then,
for any neighborhood $U$ of the boundary of $O$ there exists a horofunction
$j\in\partial G$ such that its associated horoball $H$ in $G$ satisfies

$O\cap\bigcap_{g\in H}T_{g}^{-1}(U)\neq\emptyset.$

Equivalently, 3.1 states that for any closed invariant subset $F\neq X$, which
is not isolated (i.e., it is not open), there exists a point outside of $F$
which remains close to $F$ under the action along a horoball $H$.

Notice that the distance (or even the semi-distance) on the group $G$ can be
chosen independently of the action. The theorem is worthwhile for a distance
where any horoball is large, like in the $\ell_{1}$ distance in a finitely
generated group.

The idea of the proof for $G={\mathbb{Z}}$ is as follows. By contradiction,
assume no backward or forward orbit stays in a neighborhood of the border of
$O$. A compactness argument provides a finitary version as follows: there
exists a constant $M>0$ such that no point in $O$ can have $M$ consecutive
iterates staying close to the border $\partial O$. Since $O$ is not closed and
invariant, any point close enough to the border $\partial O$ will then
contradict the constant $M$. The proof for the general case is similar but by
considering the compact set of horoballs instead of the half-spaces
$(0,+\infty)$ or $(-\infty,0)$. The geometrical fact that the horoballs are
locally tangent to a ball at the identity (2.3) helps to go from the finitary
version to the infinite one.

###### Proof.

Observe that the claim is obvious if $G$ admits an empty horoball. So we
assume that all horoballs are nonempty. This allows defining the non-negative
constant $d_{0}$ by

$d_{0}=\inf_{x\in O,H\in{\mathcal{H}}}\sup_{g\in H}{\rm
dist}(T_{g}(x),\partial O).$

Our goal is to show that $d_{0}=0$. For the sake of contradiction, assume that
$d_{0}>0$. Since the action is continuous and $B_{1}(1_{G})$ is precompact,
the family of maps $\\{T_{h}:h\in B_{1}(1_{G})\\}$ is relatively compact and
hence there exists $\delta_{0}>0$ such that ${\rm dist}(x,y)<\delta_{0}$
implies that ${\rm dist}(T_{h}(x),T_{h}(y))<d_{0}/2$ for all $h\in
B_{1}(1_{G})$.

For any $0<\epsilon<\min\\{d_{0},\delta_{0},1\\}$ and $N$ the integer given by
2.2, set $M_{\epsilon}$ to be the constant

$\begin{split}M_{\epsilon}=\sup\bigl{\\{}n\geq N:\exists
H\in{\mathcal{H}},\exists x\in X\textrm{ with }{{\rm dist}}(x,\partial
O)\geq\epsilon,\textrm{ s.t. }\forall g\in H\cap{B_{n}(1_{G})},\\\ \exists
h\in B_{\epsilon}(1_{G}),~{}{\rm dist}(T_{hg}(x),\partial
O)<\delta_{0}\bigr{\\}}.\end{split}$

By continuity of the action, the set from which we take the supremum is not
empty for any $\epsilon>0$ small enough. Moreover, we claim that
$M_{\epsilon}$ is finite. Indeed, if this is not the case, then for infinitely
many integers $n$ we may find $H_{n}\in{\mathcal{H}}$ and $x_{n}\in X$ such
that ${\rm dist}(x_{n},\partial O)\geq\epsilon$, and for all $g\in H_{n}\cap
B_{n}(1_{G})$, ${\rm dist}(T_{hg}(x_{n}),\partial O)<\delta_{0}$ for some
$h\in B_{\epsilon}(1_{G})$. Since $\partial O$ is invariant by the action,
${\rm dist}(T_{g}(x_{n}),\partial O)<d_{0}/2$ for all $g\in H_{n}\cap
B_{n}(1_{G})$. We may write $H_{n}=\\{j_{n}<0\\}$ and, taking a subsequence if
needed, we can assume $x_{n}$ and $j_{n}$ converge to some $x$ and $j$
respectively as $n\to\infty$. Let $H=\\{j<0\\}$ and take $g\in H$. For $n$
large enough we have that $g\in H_{n}\cap B_{n}(1_{G})$ and therefore ${\rm
dist}(T_{g}(x_{n}),\partial O)<d_{0}/2$. This implies that ${\rm
dist}(T_{g}(x),\partial O)\leq d_{0}/2$. This inequality is in conflict with
the definition of $d_{0}$. We conclude that $M_{\epsilon}<\infty$.

The continuity of the action and the compactness of $\partial O$ provide the
existence of $0<\delta<\epsilon$ such that

${\rm dist}(x,\partial O)<\delta~{}\text{ implies that
}\sup_{g\in\overline{B_{n_{0}}(1_{G})}}{\rm dist}(T_{g}(x),\partial
O)<\epsilon,$

where $n_{0}$ is the integer given by 2.3 associated with the constants
$M=M_{\epsilon}+1$ and $\epsilon$.

We will show by contradiction that any point $x$ such that ${\rm
dist}(x,\partial O)<\delta$ satisfies ${\rm dist}(T_{g}(x),\partial
O)<\epsilon$ for any $g\in G$. Otherwise, assume that the set

(2) $\\{d(g,1_{G})\colon g\in G\textrm{ s.t. }{\rm dist}(T_{g}(x),\partial
O)\geq\epsilon\\}$

is nonempty for some $x$ with ${\rm dist}(x,\partial O)<\delta$. Let $g_{*}\in
G$ be an element realizing the infimum of this closed set. By definition of
$\delta$, $d(g_{*},1_{G})>n_{0}$ and 2.3 provides a horoball
$H\in{\mathcal{H}}$ such that

$H\cap B_{M_{\epsilon}+1}(1_{G})\subseteq
B_{\epsilon}(1_{G})B_{d(g_{*},1_{G})}(g_{*}^{-1})=B_{\epsilon}(1_{G})B_{d(g_{*},1_{G})}(1_{G})g_{*}^{-1}.$

Thus, for any $h\in H\cap B_{M_{\epsilon}+1}(1_{G})$ ($\neq\emptyset$ by 2.2)
there exists $h_{\epsilon}\in B_{\epsilon}(1_{G})$ such that
$g=h_{\epsilon}hg_{*}$ belongs to $B_{d(g_{*},1_{G})}(1_{G})$. Since
$d(g_{*},1_{G})$ is the infimum of the set in (2), we have that ${\rm
dist}(T_{g}(x),\partial O)<\epsilon$. Hence, we get that ${\rm
dist}(T_{h_{\epsilon}h}(T_{g_{*}}(x)),\partial O)<\epsilon$ for any $h\in
H\cap B_{M_{\epsilon}+1}(1_{G})$. The definition of $M_{\epsilon}$ imposes
that ${\rm dist}(T_{g_{*}}(x),\partial O)<\epsilon$, contradicting the
definition of $g_{*}$.

Hence, for any $x\in O$ with ${\rm dist}(x,\partial O)<\delta$ (such an $x$
exists because $O$ is not closed) we get ${\rm dist}(T_{g}(x),\partial
O)<\epsilon<d_{0}$ for any $g\in G$, which is in contradiction with the
definition of $d_{0}$. ∎

Remark that in 3.1, the horoball $H$ depends on $O$ (and $U$). It is natural
to wonder if in this theorem one could fix the horoball $H$ beforehand and
show that $O\cap\bigcap_{g\in H}T_{g}^{-1}(U)\neq\emptyset$. This is not the
case, even for $G={\mathbb{Z}}$.

######  Example 3.2.

Consider the dynamical system given by a ${\mathbb{Z}}$-action of the time-1
map of a gradient system with only two fixed points on the circle. It is a
homeomorphism with two fixed points, one (the south point) is attracting and
the other (the north point $N$) is repulsive. Taking $O$ to be the complement
of the north point (and $U$ a small open ball containing it), no accumulation
point for forward orbits can be in the neighborhood of the north point.

Actually, Example 3.2 illustrates the main obstruction to finding a horoball
that satisfies the conclusion of Robinson Crusoe theorem, within a prescribed
set of horoballs: there is a repulsion property. We will show that without the
existence of repulsive sets, it is possible to restrict a priori the possible
directions of the horoball. 3.4 will provide a directed version of Robinson
Crusoe theorem.

To be more precise, we introduce the notion of repulsion of a set for a
semigroup. For a set $S\subseteq G$, we let $\langle S\rangle_{+}$ denote the
semigroup generated by the elements of $S$, i.e., the set of elements of the
form $s_{1}\cdots s_{n}$, where each $s_{i}$ belongs to $S\cup\\{1_{G}\\}$.

######  Definition 3.3.

Let $(X,T,G)$ be a topological dynamical system, $S\subseteq G$, and
$O\subseteq X$ an open $G$-invariant subset. We say that $O$ is pointwise
$S$-repulsed by its border if there exists $\delta>0$ such that for any finite
set $F\subseteq O$,

${\rm dist}(T_{g}F,\partial O)\leq\delta\textrm{ for at most finitely many
}g\in\langle S\rangle_{+}.$

This means that a finite set of points in $O$ can be separated from $\partial
O$ under the iteration of $T_{g}$ for some $g\in\langle S\rangle_{+}$. For
instance, for the north-south system in Example 3.2, the set
${\mathbb{S}}^{1}\setminus\\{N\\}$ is pointwise $\\{1\\}$-repulsed by its
border, but not pointwise $\\{-1\\}$-repulsed.

In the following, we will be interested in cases of $S$ and $O$, where this
property fails, i.e., for any $\delta>0$ there exists a finite set $F\subseteq
O$ such that for infinitely many $g\in\langle S\rangle_{+}$ there exists an
element $x_{g}\in F$ satisfying ${\rm dist}(T_{g}(x_{g}),\partial
O)\leq\delta$. In such case, we say that $O$ is not pointwise $S$-repulsed by
its border. Notice that in this case, the pigeonhole principle ensures the
existence of a point $x\in F$ such that $x_{g}=x$ for infinitely many
$g\in\langle S\rangle_{+}$.

Under this hypothesis, for some subset $S$ of $G$ with good properties, one
can state the following Robinson Crusoe theorem restricting the set of
possible horoballs.

######  Theorem 3.4 (Directed Robinson Crusoe theorem).

Let $(X,T,G)$ be a topological dynamical system where $G$ is an infinite group
with a proper right-invariant distance. Let $G_{0}\subseteq G$ be an unbounded
subset with an unbounded complement $G\setminus G_{0}$.

Assume that

  * •

$O\subsetneq X$ is an open, not closed, $G$-invariant subset of $X$.

  * •

The set $O$ is not pointwise $\tilde{S}$-repulsed by its border for some
finite subset $\tilde{S}\subseteq\cap_{H\in{\mathcal{H}}_{\partial G_{0}}}H$.

Then, for any neighborhood $U$ of the border $\partial O$ there exists a
horofunction $j\in\partial[G\setminus G_{0}^{-1}]$ such that its associated
horoball $H\in{\mathcal{H}}_{\partial[G\setminus G_{0}^{-1}]}$ satisfies

$O\cap\bigcap_{g\in H}T_{g}^{-1}(U)\neq\emptyset.$

Notice that the hypothesis of the existence of a set not pointwise
$\tilde{S}$-repulsed by its border for some finite subset $\tilde{S}$ clearly
implies restrictions on the dynamics but also algebraic restrictions on the
acting group $G$. For instance, if $G$ is a locally finite group (like
$\bigoplus{\mathbb{Z}}/2{\mathbb{Z}}$), any finitely generated semi-group
$\langle\tilde{S}\rangle_{+}$ is finite and then no $G$-action can have a set
not pointwise $\tilde{S}$-repulsed by its border.

The proof follows the same strategy as that of 3.1 but taking into
consideration the restriction on the possible set of horoballs. Here the
passage from the finite to the infinite version is ensured by the directed
geometrical 2.4. 2.5 enables us to get a large enough set of iterations that
stay close to the border, thus getting a contradiction.

###### Proof.

To simplify the notations, we set
${\mathcal{H}}_{0}={\mathcal{H}}_{\partial[G\setminus G_{0}^{-1}]}$. Notice
that the result is obvious if there is an empty horoball. So we assume that
all horoballs in ${\mathcal{H}}_{0}$ are nonempty. A similar contradiction
argument (we leave the details to the interested reader) as in 2.2, shows that
there exists an integer $N>0$ such that $H\cap B_{N}(1_{G})\neq\emptyset$ for
every horoball $H\in{\mathcal{H}}_{0}$. Let $d_{0}$ be the non-negative
constant

$d_{0}=\inf_{x\in O,H\in{\mathcal{H}}_{0}}\sup_{g\in H}{\rm
dist}(T_{g}(x),\partial O).$

For the sake of contradiction, assume that $d_{0}>0$. Since the action is
continuous and $B_{1}(1_{G})$ is precompact, the family of maps $\\{T_{g}:g\in
B_{1}(1_{G})\\}$ is relatively compact and hence there exists $\delta_{0}>0$
such that ${\rm dist}(x,y)<\delta_{0}$ implies that ${\rm
dist}(T_{g}(x),T_{g}(y))<d_{0}/2$ for all $g\in B_{1}(1_{G})$.

For any $0<\epsilon<\min(\delta_{0},1)$, set $M_{\epsilon}$ to be the constant

$\begin{split}M_{\epsilon}=\sup\bigl{\\{}n\geq N:\exists
H\in{\mathcal{H}}_{0},x\in X\textrm{ s.t. }{{\rm dist}}(x,\partial
O)\geq\epsilon,\forall g\in H\cap{B_{n}(1_{G})},\\\ \exists h\in
B_{\epsilon}(1_{G}),~{}{\rm dist}(T_{hg}(x),\partial
O)<\delta_{0}\bigr{\\}}.\end{split}$

By continuity of the action, the set from which we take the supremum is not
empty for any small enough $\epsilon>0$. Moreover, we claim that
$M_{\epsilon}$ is finite. Indeed, if this is not the case, then for infinitely
many integers $n$ we may find $H_{n}\in{\mathcal{H}}_{0}$ and $x_{n}\in X$
such that ${\rm dist}(x_{n},\partial O)\geq\epsilon$, and for all $g\in
H_{n}\cap B_{n}(1_{G})$, ${\rm dist}(T_{hg}(x_{n}),\partial O)<\delta_{0}$ for
some $h\in B_{\epsilon}(1_{G})$. It follows that ${\rm
dist}(T_{g}(x_{n}),\partial O)<d_{0}/2$ for all $g\in H_{n}\cap B_{n}(1_{G})$.
We may write $H_{n}=\\{j_{n}<0\\}$ and taking a subsequence if needed, we can
assume that $x_{n}$ converges to some $x$ and $j_{n}$ to some $j$. Then the
horoball $H=\\{j<0\\}$ belongs to ${\mathcal{H}}_{0}$. Take $g\in H$. For $n$
large enough we have that $g\in H_{n}\cap B_{n}(1_{G})$ and therefore ${\rm
dist}(T_{g}(x_{n}),\partial O)<d_{0}/2$. This implies that ${\rm
dist}(T_{g}(x),\partial O)\leq d_{0}/2$. This inequality is in conflict with
the definition of $d_{0}$. We conclude that $M_{\epsilon}<\infty$.

Let $n_{0}$ be the integer given by 2.4 associated with the constants
$M=M_{\epsilon}+1$, $\epsilon$ and the set $A=G\setminus G_{0}$. Let
$S=\tilde{S}^{-1}$, that is, a finite set in $\\{g\in G:\forall j\in\partial
G_{0},\ j(g^{-1})<0\\}$ such that $O$ is not pointwise $S^{-1}$-repulsed by
its border. Set $\eta$ to be $-\sup_{s\in S,j\in\partial G_{0}}j(s^{-1})>0$.
Let $n_{1}$ be a constant provided by 2.5 for each element $g$ in the finite
set $S$. Furthermore, taking $n_{1}$ large enough, we may assume
$\overline{B_{n_{0}}(s)}\subset B_{n_{1}}(1_{G})$ for any element $s\in S$.

The continuity of the action and the compactness and invariance of $\partial
O$ provide the existence of $0<\delta<\epsilon$ such that

${\rm dist}(x,\partial O)<\delta~{}\text{ implies that
}\sup_{g\in\overline{B_{n_{1}}(1_{G})}}{\rm dist}(T_{g}(x),\partial
O)<\epsilon.$

We will show that there is a set $C\subseteq G$ containing arbitrary large
balls such that ${\rm dist}(T_{g}(x),\partial O)<\epsilon$ for any $g\in C$
and for any point $x\in O$ satisfying ${\rm dist}(x,\partial O)<\delta$. To
prove it, consider for a point $x$ at distance less than $\delta$ to $\partial
O$, the set

$C(x)=\\{g\in G:{\rm dist}(T_{g}(x),\partial O)<\epsilon\\}.$

By the choice of $\delta$, this set $C(x)$ contains the ball
$B_{n_{1}}(1_{G})$. First, we claim it satisfies the following property:

(P) $\displaystyle\textrm{If for }g\in G,\exists H\in{\mathcal{H}}_{0}\textrm{
s.t. }\left[H\cap B_{M_{\epsilon}+1}(1_{G})\right]g\subseteq
B_{\epsilon}(1_{G})C(x),\textrm{ then }g\in C(x).$

This comes from the fact that if for any $h$ in $H\cap
B_{M_{\epsilon}+1}(1_{G})$ (nonempty by the definition of $N$) there is
${h}_{\epsilon}\in\left(B_{\epsilon}(1_{G})\right)^{-1}$ such that ${\rm
dist}(T_{{h}_{\epsilon}h}(T_{g}x),\partial O)<\epsilon$. The definition of
$M_{\epsilon}$ implies then that ${\rm dist}(T_{g}(x),\partial O)<\epsilon$,
i.e., $g\in C(x)$.

Assume that $C(x)$ contains a ball $B_{R}(\bar{g})$ of radius $R\geq n_{1}$.
We claim that for any $s\in S$ the set $C(x)$ also contains the ball of
greater radius $B_{R+\eta}(s\bar{g})$. Otherwise, there is $h^{*}\in
B_{R+\eta}(s\bar{g})\setminus C(x)$ minimizing the distance
$d(h^{*},s\bar{g})$. This minimality condition implies
$B_{d(h^{*},s\bar{g})}(s\bar{g})\subset C(x)$. Also, notice that by the choice
of the constant $n_{1}$, the ball $\overline{B_{n_{0}}(s\bar{g})}$ is included
in $B_{R}(\bar{g})$, hence in $C(x)$. So we have $d(h^{*},s\bar{g})>n_{0}$. If
$h^{*}$ belongs to $[G\setminus G_{0}]s\bar{g}$, 2.4 ensures the existence of
a horoball $H\in{\mathcal{H}}_{0}$ such that

$\left[H\cap B_{M_{\epsilon}+1}(1_{G})\right]h^{*}\subseteq
B_{\epsilon}(1_{G})B_{d(h^{*},s\bar{g})}(s\bar{g})\subseteq
B_{\epsilon}(1_{G})C(x).$

This is impossible by Property (P). It follows that $h^{*}$ belongs to
$G_{0}s\bar{g}$. Then, 2.5 applied to $g=s$, implies that $h^{*}$ belongs to
$B_{R}(\bar{g})\subseteq C(x)$: again a contradiction.

Applying inductively the former claim, we get for any sequence $(s_{n})_{n\geq
0}$ in $S$ that $C(x)$ contains the set $\bigcup_{n\geq
0}B_{n_{1}+n\eta}(s_{n}\cdots s_{1})$, which contains arbitrarily large balls.
Therefore $C(x)$ contains the set $C=\bigcup_{n\geq
0}\bigcup_{s_{1},\ldots,s_{n}\in S}B_{n_{1}+n\eta}(s_{n}\cdots s_{1})$. Hence
any $x\in O$ such that ${\rm dist}(x,\partial O)<\delta$, satisfies

${\rm dist}(T_{g}(x),\partial O)<\epsilon\ \text{ for all }g\in C.$

Given that the set $O$ is not pointwise $S^{-1}$-repulsed by its border, there
exists $x\in O$ such that ${\rm dist}(T_{\tilde{g}}(x),\partial O)<\delta$ for
infinitely many $\tilde{g}\in\langle S^{-1}\rangle_{+}$. For any such
$\tilde{g}$, it follows that ${\rm dist}(T_{g}(x),\partial O)<\epsilon$,
$\forall g\in C\tilde{g}$. Since any $\tilde{g}$ is of the form $(s_{n}\cdots
s_{1})^{-1}$, with $s_{i}\in S$, we get that ${\rm dist}(T_{h}(x),\partial
O)<\epsilon$ for any $h$ in $B_{n_{1}+n\eta}(1_{G})$. Since this is true for
infinitely many $\tilde{g}$, this remains true for infinitely many integers
$n$, hence for any $h$ in $G$. This contradicts the definition of $d_{0}$. ∎

##  4\. Extensions of Schwartzman theorem

In this section we apply the Robinson Crusoe theorems to get several
extensions of a classical result due to Schwartzman in his Ph.D. Thesis [28]
to general group actions. Recall that Schwartzman’s theorem states that given
a homeomorphism $T\colon X\to X$ of an infinite compact metric space $X$ and
$\epsilon>0$ there exist positive (resp. negative) $\epsilon$-asymptotic
pairs. That is, different points $x,y\in X$ such that ${\rm
dist}(T^{n}x,T^{n}y)\leq\epsilon$ for all $n\geq 0$ (resp. $n\leq 0$). This is
equivalent to say that there is no positively (or negatively) expansive
homeomorphisms on an infinite compact space. This result can be seen as a
version in topological dynamics of the classical Morse-Hedlund’s theorem on
the complexity of an infinite subshift [24].

A crucial issue in extending Schwartzman’s theorem to general group actions is
to define a good notion of asymptoticity that extends the one of asymptotic
pair for ${\mathbb{Z}}$-symbolic systems. We propose a notion using the
elements of geometric group theory that we developed in the last sections. We
recall that a factor map between topological dynamical systems $(X,T,G)$ and
$(Y,S,G)$ is a continuous and onto map $\pi\colon X\to Y$ such that
$S_{g}\circ\pi=\pi\circ T_{g}$ for all $g\in G$.

######  Definition 4.1.

For two dynamical systems $(X,T,G)$ and $(Y,S,G)$, a factor map $\pi\colon
X\to Y$, $\epsilon>0$ and a horoball $H$, we call an $(\epsilon,H)$-asymptotic
pair relative to $\pi$ any pair $(x,y)$ of two different points $x,y\in X$
such that $\pi(x)=\pi(y)$ and ${\rm dist}(T_{g}x,T_{g}y)\leq\epsilon$ for all
$g\in H$.

When the system $Y$ is trivial we simply say that $(x,y)$ is an
$(\epsilon,H)$-asymptotic pair.

We might naively expect from Schwartzman’s theorem that any group action
admits $(\epsilon,H)$-asymptotic pairs for each horoball $H$. But this is
false even for ${\mathbb{Z}}^{2}$-actions, as shown by Ledrappier’s shift
example.

######  Example 4.2.

(Ledrappier’s example) Consider the ${\mathbb{Z}}^{2}$ shift

$X=\\{(x_{i,j})_{(i,j)\in{\mathbb{Z}}^{2}}\in({\mathbb{Z}}/2{\mathbb{Z}})^{{\mathbb{Z}}^{2}}:x_{i,j}+x_{i+1,j}+x_{i,j+1}=0\mod
2\quad\forall i,j\in{\mathbb{Z}}\\},$

and the ${\mathbb{Z}}^{2}$-action generated by the horizontal and vertical
shifts. If $v\in{\mathbb{R}}^{2}$ is not an outgoing vector normal to a face
of the unit simplex in ${\mathbb{R}}^{2}$, then for each $x\in X$, the
coordinates of $x$ within $H_{v}=\\{z:\langle z,v\rangle<0\\}$ determine all
the coordinates of $x$. So, there is no $(\epsilon,H_{v})$-asymptotic pair for
any small enough $\epsilon>0$. Conversely, for each of the three normal
outgoing vectors $v$ of the face of the unit simplex, there is an asymptotic
pair for the half-space $H_{v}$. See [7, Example 2.7] for details.

Nevertheless, an important step in extending Schwartzman’s theorem to
${\mathbb{Z}}^{d}$-actions has been done by Boyle and Lind through the notion
of nonexpansive direction [7]. In contrast with the ${\mathbb{Z}}$-case, their
result provides the existence of $(\epsilon,H)$-asymptotic pairs only for some
horoball $H$. Technically, it ensures the existence of some nonexpansive
directions from which the former claim can be easily deduced.

###  4.1. Existence of $(\epsilon,H)$-asymptotic pairs

Thanks to 3.1 we propose a first extension of Schwartzman’s theorem (more in
the spirit of Boyle and Lind’s theorem) to any countable group and more
generally to any second countable and locally compact group. Moreover, our
extension is relative to a factor map.

A factor map $\pi$ is bounded to one (resp. constant to one) if the
cardinality of the $\pi$-fibers is uniformly bounded (resp. equals to a
constant).

######  Theorem 4.3.

Let $(X,T,G)$ and $(Y,S,G)$ be topological dynamical systems, where $G$ is an
infinite group with a proper right-invariant distance, and let $\pi\colon X\to
Y$ be a factor map. Then, either

  * •

$\pi$ is bounded to one,

  * •

or for any $\epsilon>0$ there exist a horoball $H\in{\mathcal{H}}$ and an
$(\epsilon,H)$-asymptotic pair $(x,y)\in X^{2}$ relative to $\pi$.

In particular, when $(Y,S,G)$ is trivial and $X$ is infinite, there exists an
$(\epsilon,H)$-asymptotic pair $(x,y)\in X^{2}$.

Note that in the case of a trivial $(Y,S,G)$, 4.3 recovers 1.1 announced in
the introduction.

Interestingly, in the context of symbolic dynamics, 4.3 provides the existence
of an asymptotic pair for any infinite subshift. More precisely, for an
infinite closed subset $X$ of $A^{G}$ (where $A$ is a finite set and $G$ is a
countable group), invariant by the natural shift action $T$ induced by
translations on $G$, there exists a horoball $H$ and two points $x\neq y\in X$
that coincide when restricted to $H$.

Remark that 4.3 does not say that its items are mutually exclusive. In fact,
many situations can arise: there exist factor maps that are bounded to one but
do not identify asymptotic pairs (e.g., consider an irrational rotation on the
complex circle with the factor map $z\mapsto z^{n}$). On the other hand, an
example of an unbounded to one factor map identifying asymptotic points is
also given by a dynamic on the circle. For instance, consider a Denjoy’s
homeomorphism. It preserves a Cantor set $C$. It is standard that it has
positively asymptotic pairs: namely the extreme points of a connected
component of the complementary of $C$. Moreover, the homeomorphism factorizes
onto an irrational rotation by a factor map identifying these connected
components. Finally, recall that there exist Denjoy’s examples where the
restrictions on the Cantor set $C$ provide factor maps that are both bounded
to one and identify the asymptotic pairs.

Let us stress that 4.3 is intrinsic to the system and does not depend on the
embedding into another space with some constraints (like the dimension,
connectivity, …). This differs, for instance, from what the usual study of
minimal sets of homeomorphisms (like the one of Denjoy homeomorphisms) would
suggest.

###### Proof.

Consider the system $(R_{\pi},T^{(2)},G)$, where $R_{\pi}=\\{(x,y)\in X\times
X:\pi(x)=\pi(y)\\}$ and $T^{(2)}$ is the diagonal action
$T^{(2)}_{g}(x,y)=(T_{g}(x),T_{g}(y))$. Let $O=R_{\pi}\setminus\Delta_{X}$,
where $\Delta_{X}$ is the diagonal on $X$. Clearly, $O$ is open and
$G$-invariant. If $O$ is closed, then there exists $\delta>0$ such that ${\rm
dist}(x,y)\geq\delta$ for any $(x,y)\in O$. Then the compactness of $X$
implies that $\pi$ is bounded to one.

Now assume $O$ is not closed. This implies that $\emptyset\neq\partial
O\subseteq\Delta_{X}$. For any $\epsilon>0$, set $U=\\{(x,y)\colon{\rm
dist}(x,y)<\epsilon\\}$ an open neighborhood of $\partial O$. By 3.1, for any
$\epsilon>0$ there exist a horoball $H$ in ${\mathcal{H}}$ and different
points $x,y\in X$ with $\pi(x)=\pi(y)$ such that for all $g\in H$ we have
${\rm dist}(T_{g}(x),T_{g}(y))<\epsilon$. ∎

Similarly, the application of the Directed Robinson Crusoe theorem provides a
directed version of 4.3. This is a second extension of Schwartzman’s theorem.
For expository reasons, we state a version which is valid for groups admitting
non periodic elements and where the inverse map is an isometry, i.e.,
$d(x^{-1},y^{-1})=d(x,y)$. This is the case, for instance, for the $\ell_{1}$
distance of a finitely generated group. The inverse map is also an isometry
for any distance $d$ that is bi-invariant, i.e., that is both left and right-
invariant: $d(gxh,gyh)=d(x,y)$ for any $g,h,x,y\in G$. This occurs for
instance for abelian groups.

######  Theorem 4.4.

Let $\pi\colon X\to Y$ be a factor map between the topological dynamical
systems $(X,T,G)$ and $(Y,S,G)$, where $G$ is an infinite group with a proper
right-invariant distance and where the inverse map is an isometry. Assume that
$\pi$ is not bounded to one. Let $k\in G$ be an element generating an
unbounded subgroup $\langle k\rangle$. Then, for any $\epsilon>0$, there
exists a horoball $H\in{\mathcal{H}}$ such that

  * •

the horoball $H$ does not contain $k$;

  * •

there exists an $(\epsilon,H)$-asymptotic pair $(x,y)$ relative to $\pi$.

Observe that the first item implies a strong restriction on the horoballs. For
instance, if $G={\mathbb{Z}}^{2}$, the set of $\ell_{2}$ horoballs that do not
contain a given point $k\in{\mathbb{Z}}^{2}$ forms a half circle of directions
when identified with their normal outgoing vectors (see Section 5 for further
discussion).

Also, notice that following the proof (given below), we can relax the
hypothesis on the isometric action of the inverse map by only considering
right-invariant distances. But the price to pay is to get the following more
intricate statement.

######  Theorem 4.5 (4.4 general statement).

Let $\pi\colon X\to Y$ be a factor map between the topological dynamical
systems $(X,T,G)$ and $(Y,S,G)$, where $G$ is an infinite group with a proper
right-invariant distance. Assume that $\pi$ is not bounded to one. Let $k\in
G$ be an element generating an unbounded subgroup $\langle k\rangle$. Then,
for any $\epsilon>0$, there exists an unbounded sequence
$(g_{n})_{n\in{\mathbb{N}}}$ in $G$ such that

  1. (1)

the horoball $\displaystyle
H^{\prime}=\\{\lim_{n\to\infty}b_{g_{n}^{-1}}<0\\}$ does not contain $k$;

  2. (2)

the horoball $\displaystyle H=\\{\lim_{n\to\infty}b_{g_{n}}<0\\}$ admits an
$(\epsilon,H)$-asymptotic pair $(x,y)$ relative to $\pi$.

Actually, we believe the correct general setting on the distance to get a
similar conclusion is that the inverse map $g\mapsto g^{-1}$ defines a coarse
map (see [10]). But the statement would be unnecessarily complicated for our
purpose.

###### Proof of 4.4.

The proof starts similarly to that of 4.3. We keep the same notations: i.e.,
we consider the system $(R_{\pi},T^{(2)},G)$, where $R_{\pi}=\\{(x,y)\in
X\times X:\pi(x)=\pi(y)\\}$ and $T^{(2)}$ is the diagonal action
$T^{(2)}_{g}(x,y)=(T_{g}(x),T_{g}(y))$. The set $O$ denotes
$R_{\pi}\setminus\Delta_{X}$, where $\Delta_{X}$ is the diagonal in $X\times
X$. The set $O$ is open and $G$-invariant and the same proof as in 4.3 shows
it is not closed.

Moreover, for any $\delta>0$ there exists $\epsilon>0$ such that if $(x,y)\in
O$ and ${\rm dist}(x,y)\leq\epsilon$ then ${\rm dist}((x,y),\partial
O)\leq\delta$. Indeed, for the sake of a contradiction, assume this is not
true. Taking a subsequence if needed, we may consider a sequence
$((x_{n},y_{n}))_{n\in{\mathbb{N}}}$ in $O$ with $(x_{n},y_{n})\to(z,z)$ when
$n\to\infty$ for some $z\in X$ and ${\rm dist}((x_{n},y_{n}),\partial
O)>\delta$ for all $n\in{\mathbb{N}}$. But, by definition, $(z,z)\in\partial
O$, which contradicts the fact that ${\rm dist}((x_{n},y_{n}),\partial
O)>\delta$.

Since the group $\langle k\rangle$ is unbounded, both semigroups $\langle
k\rangle_{+}$ and $\langle k^{-1}\rangle_{+}$ are unbounded. We claim that $O$
is not pointwise $\\{k^{-1}\\}$-repulsed by its border. Indeed, for $\delta>0$
let $\epsilon>0$ be such that $(x,y)\in O$ and $d(x,y)\leq\epsilon$ implies
$d((x,y),\partial O)\leq\delta$. Let $B_{1},\ldots,B_{m}$ be distinct open
balls of diameter $\epsilon/2$ whose union covers $X$. Let
$x_{1},\ldots,x_{m+1}$ be distinct points with
$\pi(x_{1})=\pi(x_{2})=\cdots=\pi(x_{m+1})$. Let $F=\\{(x_{i},x_{j}):i\neq
j\\}\subseteq O$. Then, by the pigeonhole principle, for any $g\in\langle
k^{-1}\rangle_{+}\subseteq G$ there exist $i$ and $j$ such that $T_{g}(x_{i})$
and $T_{g}(x_{j})$ belong to a same ball $B_{k}$. This means that ${\rm
dist}(T_{g}(x_{i}),T_{g}(x_{j}))\leq\epsilon$, which implies that
$(T_{g}(x_{i}),T_{g}(x_{j}))$ is $\delta$-close to $\partial O$. The claim is
proved.

So the set $O$ is non pointwise $\\{k^{-1}\\}$-repulsed by its border. Set
$G_{k}=\\{g\in G:k^{-1}\in B_{d(1,g)}(g)\\}$. It can be checked that $G_{k}$
and $G\setminus G_{k}$ are unbounded since they contain unbounded subsets of
$\langle k^{-1}\rangle_{+}$ and $\langle k\rangle_{+}$ respectively. Moreover,
by definition $k^{-1}$ belongs to $\bigcap_{H\in{\mathcal{H}}_{\partial
G_{k}}}H$. By the directed Robinson Crusoe theorem 3.4 applied to $O$ and
$G_{0}=G_{k}$, for any $\epsilon>0$ there exist a horoball $H$ in
${\mathcal{H}}_{\partial[G\setminus G_{k}^{-1}]}$ and different points $x,y\in
X$ with $\pi(x)=\pi(y)$ such that for all $g\in H$ we have ${\rm
dist}(T_{g}(x),T_{g}(y))<\epsilon$.

Write $\displaystyle H=\\{\lim_{n\to\infty}b_{g_{n}}<0\\}$ for a sequence
$(g_{n})_{n\in{\mathbb{N}}}$ in $G\setminus G_{k}^{-1}$ going to infinity.
Since $g_{n}^{-1}\notin G_{k}$ we have that for all $n\in{\mathbb{N}}$,
$k^{-1}\not\in B_{d(1,g_{n})}(g_{n}^{-1})$. Since the inverse map is an
isometry, this means that $k\not\in B_{d(1,g_{n})}(g_{n})$. The proof is
complete. ∎

The Robinson Crusoe theorems are valid for second countable, locally compact
groups (like ${\mathbb{R}}^{d}$ or Lie groups). But in some cases, they hold
for trivial reasons. For instance, in the case of a ${\mathbb{R}}^{d}$-flow,
the notion of $(\epsilon,H)$-asymptotic pair is trivial: any non fixed point
with any small translation of it by the flow give an asymptotic pair. This
problem already occurs in the definition of expansive flows (see [6]). This
leads to the following question:

######  Question 4.6.

Provide nontrivial versions of 4.3 and 4.4 for flows of (connected) second
countable, locally compact groups actions.

###  4.2. Structure of the set of asymptotic pairs

The structure of the set of pairs that are $(\epsilon,H)$-asymptotic for some
horoball $H$ and $\epsilon>0$ depends very much on the particular dynamical
system we are working with. It can be really small, for instance finite, and
even restricted to only two orbits (e.g., Sturmian subshifts, Denjoy examples
of homeomorphism of the circle). Hence it does not form, in general, a generic
set in the topological or measurable frameworks. At the opposite, it can also
be really large (e.g., in the fullshift there is a Cantor set of such pairs).

However, in some cases, the asymptotic pairs reflect features of the global
structure of the ambient space. For instance, for a product dynamical system,
it is simple to check that the set of asymptotic pairs also has a product
structure. Less directly, we have the following result for spaces with
nontrivial connected components.

######  Corollary 4.7.

Let $(X,T,G)$ be a topological dynamical system with $\dim X>0$ and $G$ be an
infinite group with a proper right-invariant distance. Then, for any
$\epsilon>0$ there exist a horoball $H$ in $G$ and a connected set $F\subseteq
X$ such that $0<\textrm{diam}(T_{g}(F))<\epsilon$ for every $g\in
H\cup\\{1_{G}\\}$.

In particular, this result shows, in the case of a dynamic on a connected set,
that it admits a (stable) set

$W^{s}_{\epsilon,H}(x)=\\{y\in X:\forall h\in H,\ {\rm
dist}(T_{h}(x),T_{h}(y))<\epsilon\\}$

of positive topological dimension for some $x\in X$ and some horoball
$H\in{\mathcal{H}}$. In differentiable dynamics, this leads to the meaningful
notion of stable manifold. Let us also point out that this result is the
starting point in Mañé’s proof that the minimal components of expansive
${\mathbb{Z}}$-dynamics are zero-dimensional [21].

###### Proof.

Let ${\mathcal{X}}$ denote the set of all closed connected subsets of $X$.
Since $X$ has positive dimension, basics on dimension theory imply that
${\mathcal{X}}$ is nonempty. It is also classical that ${\mathcal{X}}$ is
compact for the Hausdorff metric. Let $O\subseteq{\mathcal{X}}$ denote the set
of positive diameter connected subsets of $X$. It is clearly invariant, open
and non closed. Then, 3.1 gives us immediately the desired conclusion. ∎

##  5\. Set of nonexpansive horoballs for a ${\mathbb{Z}}^{d}$-action

In Section 4 we have shown the existence of horoballs admitting
$\epsilon$-asymptotic pairs. Here we try to obtain more information about the
set of such horoballs. For this we follow the same path initiated by Boyle and
Lind in [7]. In their work the authors consider the notion of nonexpansive
direction which corresponds in our context to the border of horoballs
admitting $\epsilon$-asymptotic pairs for the $\ell_{2}$ distance in
${\mathbb{Z}}^{d}$. In this case, horoballs are open half-spaces and their
border are hyperplanes, i.e., directions for $d=2$. Remark that by only
considering the border of a half-space, we lose information about its
orientation, as the half-space and its opposite share the same border. This is
an important difference between our approach and that of nonexpansive
directions.

Boyle and Lind showed in [7] that the set of nonexpansive directions (under
their definition) is a closed subset within the set of all the directions.
Furthermore, their work, together with Hochmans’s [17] shows that any nonempty
closed set can be realized as the set of nonexpansive direction of some
${\mathbb{Z}}^{2}$-action.

In this section we restrict ourselves to ${\mathbb{Z}}^{d}$-actions and we
propose a notion of nonexpansive horoballs (see 5.1, similar to the one of
[30]). The result of [7] about closeness of the set of nonexpansive directions
can be easily adapted to the context of $\ell_{2}$ horoballs. We use this fact
to obtain a restriction on the set of nonexpansive horoballs of a
${\mathbb{Z}}^{d}$-action: the intersection of all its nonexpansive horoballs
has to be empty (5.6). This is an extension of Schwartzman’s result.

######  Definition 5.1.

For dynamical systems $(X,T,G)$ and $(Y,S,G)$, a factor map $\pi\colon X\to Y$
and $\epsilon>0$, a horoball $H$ is said to be $(\epsilon,\pi)$-nonexpansive
if there exist two different points $x,y\in X$ such that $\pi(x)=\pi(y)$ and
${\rm dist}(T_{g}x,T_{g}y)\leq\epsilon$ for all $g\in H$ (i.e., $(x,y)$ is an
$(\epsilon,H)$-asymptotic pair relative to $\pi$).

When $Y$ is the trivial system, we simply call the horoball
$\epsilon$-nonexpansive.

At the opposite, a horoball $H$ that is not $(\epsilon,\pi)$-nonexpansive,
satisfies the following property: for every $x,y$ such that $\pi(x)=\pi(y)$,

$\sup_{g\in H}{\rm dist}(T_{g}(x),T_{g}(y))\leq\epsilon\text{ implies }x=y.$

In this sense, the horoball $H$ determines the points of $X$. For this reason,
C. Zinoviadis in [30] uses the term nondeterministic instead of nonexpansive.

For ${\mathbb{Z}}^{d}$ or ${\mathbb{R}}^{d}$-actions, we set the collection of
$(\epsilon,\pi)$-nonexpansive $\ell_{2}$ horoballs (identified with their
normal outgoing vectors) as

(3) $\displaystyle\operatorname{ND}_{\ell_{2}}(X,\epsilon,\pi)$
$\displaystyle=\\{v\in{\mathbb{S}}^{d-1}:\\{x\in{\mathbb{R}}^{d}:\langle
x,v\rangle<0\\}\textrm{ is }(\epsilon,\pi)\textrm{-nonexpansive}\\},$ (4)
$\displaystyle\operatorname{ND}_{\ell_{2}}(X,\pi)$
$\displaystyle=\bigcap_{\epsilon>0}\operatorname{ND}_{\ell_{2}}(X,\epsilon,\pi).$

Also, for the trivial factor we simply denote these sets by
$\operatorname{ND}_{\ell_{2}}(X,\epsilon)$ and
$\operatorname{ND}_{\ell_{2}}(X)$. To avoid confusion between the notions, we
refer to the elements of $\operatorname{ND}_{\ell_{2}}(X,\pi)$ (resp.
$\operatorname{ND}_{\ell_{2}}(X)$) as $\pi$-nondeterministic (resp.
nondeterministic) vectors. Similar arguments as the ones in [7, Lemma 3.4]
show the closeness of the set of nondeterministic vectors for the usual
topology on the sphere ${\mathbb{S}}^{d-1}$.

######  Lemma 5.2.

Let $(X,T,{\mathbb{Z}}^{d})$ be a dynamical system and $\pi\colon X\to Y$ a
factor map onto the system $(Y,S,{\mathbb{Z}}^{d})$. Then, for any
$\epsilon>0$ the sets $\operatorname{ND}_{\ell_{2}}(X,\epsilon,\pi)$ and
$\operatorname{ND}_{\ell_{2}}(X,\pi)$ are closed subsets of the unit sphere.

For the reader’s convenience, we provide a proof that
$\operatorname{ND}_{\ell_{2}}(X,\epsilon,\pi)$ is closed.

###### Proof.

First, we assume that for any pair of different points $x,y$ such that
$\pi(x)=\pi(y)$, there exists a $g\in{\mathbb{Z}}^{d}$ such that ${\rm
dist}(T_{g}(x),T_{g}(y))>\epsilon$. Otherwise we have
$\operatorname{ND}_{\ell_{2}}(X,\epsilon,\pi)={\mathbb{S}}^{d-1}$ and there is
nothing to prove.

For $v\in\operatorname{ND}_{\ell_{2}}(X,\epsilon,\pi)$, there are two points
$x,y$ such that $\pi(x)=\pi(y)$, and ${\rm dist}(T_{g}(x),T_{g}(y))<\epsilon$,
for any $g$ in the half-space $H_{v}=\\{g:\langle v,g\rangle<0\\}$. By the
initial assumption, we may pick $\tilde{g}$, minimizing the distance to
$H_{v}$ such that ${\rm dist}(T_{\tilde{g}}(x),T_{\tilde{g}}(y))>\epsilon$.
Hence ${\rm dist}(T_{g\tilde{g}}(x),T_{g\tilde{g}}(y))<\epsilon$ for any $g\in
H_{v}$.

We deduce the next characterization:
$v\in\operatorname{ND}_{\ell_{2}}(X,\epsilon,\pi)$ if and only if there exist
$x,y\in X$ such that, $\pi(x)=\pi(y),{\rm dist}(x,y)>\epsilon$, and ${\rm
dist}(T_{g}(x),T_{g}(y))<\epsilon$, for any $g$ such that $\langle
v,g\rangle<0$. Note that the converse part is trivial. From this
characterization, it is straightforward to check that
$\operatorname{ND}_{\ell_{2}}(X,\epsilon,\pi)$ is closed.

The closeness of $\operatorname{ND}_{\ell_{2}}(X,\pi)$ follows from that of
$\operatorname{ND}_{\ell_{2}}(X,\epsilon,\pi)$ since it is the intersection of
a nested sequence of such sets. ∎

More interestingly, 4.3 provides conditions that ensure the nonemptiness of
the set $\operatorname{ND}_{\ell_{2}}(X,\epsilon,\pi)$ for every $\epsilon>0$.
The former argument then implies that the set
$\operatorname{ND}_{\ell_{2}}(X,\pi)$ is nonempty. This shows in particular
that the horoball given by 4.3 may be taken uniformly in $\epsilon>0$ for the
$\ell_{2}$ distance. This leads to the following notion of nonexpansivity
along a subset (similar to [7]), which becomes relevant when considering
horoballs.

######  Definition 5.3.

A topological dynamical system $(X,T,G)$ with $G$ countable is said to be
expansive along a subset $H$ if there exists $\epsilon_{0}>0$ such that

$\sup_{g\in H}{\rm dist}(T_{g}x,T_{g}y)\leq\epsilon_{0}\text{ implies }x=y.$

If $H$ fails to meet this condition, the system is said to be non
$H$-expansive.

Observe that this notion is meaningless in the context of ${\mathbb{R}}^{d}$
flows since any pair of points $(x,T_{\epsilon}x)$ for small enough
$\epsilon>0$ shows that such flow is non $H$-expansive for any horoball.

With this definition 4.3 can be stated as follows (in the spirit of
Schwartzman’s result).

######  Corollary 5.4.

Let $(X,T,{\mathbb{Z}}^{d})$ be an infinite topological dynamical system.
Then, it is non $H$-expansive for some $\ell_{2}$ horoball $H$.

In addition, the closeness property together with 4.4 (the directed version on
the existence of asymptotic pairs) enables us to guarantee the existence of a
nonexpansive horoball (for ${\mathbb{Z}}^{d}$ systems) within any given half-
space.

######  Theorem 5.5.

Let $\pi\colon X\to Y$ be a factor map between the topological dynamical
systems $(X,T,{\mathbb{Z}}^{d})$ and $(Y,S,{\mathbb{Z}}^{d})$. Assume that
$\pi$ is not bounded to one. Then, any closed half-space
$H\subseteq{\mathbb{R}}^{d}$ contains a $\pi$-nondeterministic vector, i.e.,

$\operatorname{ND}_{\ell_{2}}(X,\pi)\cap H\neq\emptyset.$

When $X$ is infinite and $Y$ is the trivial factor, we obtain that any closed
half-space contains a nondeterministic vector.

###### Proof.

We endow ${\mathbb{Z}}^{d}$ with the $\ell^{2}$ distance. Let
$H\subseteq{\mathbb{R}}^{d}$ be a closed half-space and write
$H=\\{x\in{\mathbb{R}}^{d}:\langle x,c\rangle\geq 0\\}$ for some
$c\in{\mathbb{R}}^{d}$. Let $(g_{n})_{n\in{\mathbb{N}}}$ be a sequence going
to infinity such that the distance of $g_{n}$ to the ray
$\\{tc:t\in\mathbb{R}_{+}\\}$ goes to 0. Let ${\mathcal{H}}_{n}$ denote the
set of horoballs that do not contain $g_{n}$. Identifying a horoball with its
outward unit normal vector, we may identify ${\mathcal{H}}_{n}$ with the set
$S_{n}=\\{v\in{\mathbb{S}}^{d-1}:\langle g_{n},v\rangle\geq 0\\}$. Let
$\epsilon>0$. By 4.4, for each $n\in{\mathbb{N}}$ we may find a vector
$v_{n}\in\operatorname{ND}_{\ell_{2}}(X,\epsilon,\pi)\cap S_{n}$. Taking a
subsequence if needed, we may assume that $v_{n}\to v$ as $n\to\infty$. Since
$\operatorname{ND}_{\ell_{2}}(X,\epsilon,\pi)$ is closed by 5.2, we get
$v\in\operatorname{ND}_{\ell_{2}}(X,\epsilon,\pi)$. We may write
$g_{n}=t_{n}c+\epsilon_{n}$ with $\|\epsilon_{n}\|\to 0$ and
$t_{n}\nearrow\infty$ as $n\to\infty$. From $\langle v_{n},g_{n}\rangle\geq 0$
we obtain that $\langle v,c\rangle\geq 0$, and hence $v$ belongs to $H$. As
$\epsilon>0$ is arbitrary, using again 5.2, we obtain the desired conclusion.

∎

A special case of 5.5, for $d=2$ and $Y$ the trivial system, was announced by
P. Guillon, J. Kari and C. Zinoviadis in 2015 [15], but to the best of our
knowledge, the proof was not communicated to the community. An immediate
corollary of 5.5 is the following restriction on the set of nonexpansive half-
spaces.

######  Corollary 5.6.

Let $\pi\colon X\to Y$ be a factor map between the topological dynamical
systems $(X,T,{\mathbb{Z}}^{d})$ and $(Y,S,{\mathbb{Z}}^{d})$. Assume that
$\pi$ is not bounded to one. Then the origin belongs to the convex hull
generated by the elements in $\operatorname{ND}_{\ell_{2}}(X,\pi)$. In
particular, the intersection of all the $\pi$-nonexpansive $\ell_{2}$
horoballs is empty:

$\bigcap_{v\in\operatorname{ND}_{\ell_{2}}(X)}\\{y\in{\mathbb{R}}^{d}:\langle
y,v\rangle<0\\}=\emptyset.$

When $Y$ is the trivial system we obtain 1.3 announced in the introduction.
Recall that $\operatorname{ND}_{\ell_{2}}(X)$ is never empty when $X$ is
infinite (4.3). It follows from the empty intersection condition that
$\operatorname{ND}_{\ell_{2}}(X)$ always contains at least $2$ elements. More
precisely, for $d=2$, 5.6 implies the following dichotomy. Either
$\operatorname{ND}_{\ell_{2}}(X)$ contains at least $2$ independent vectors
$v_{1},v_{2}$ and a third one in the convex cone generated by $-v_{1}$ and
$-v_{2}$, or $\operatorname{ND}_{\ell_{2}}(X)$ contains only two elements
$v_{1},v_{2}$ that are opposites (i.e., $v_{1}=-v_{2}$). These cases are
optimal. For instance, for the Ledrappier’s subshift the set
$\operatorname{ND}_{\ell_{2}}(X)$ consists of the normal outgoing vectors of
the faces of a triangle (see Example 4.2). Concerning the second case, Hochman
showed in [17] that any direction in ${\mathbb{R}}^{2}$ appears as the unique
nonexpansive direction of some ${\mathbb{Z}}^{2}$-dynamical system. Since any
nonexpansive half-space defines a nondeterministic direction and by 5.6, this
system has only two symmetric nonexpansive half-spaces.

###### Proof.

By contradiction, assume that the origin does not belong to the convex hull
generated by the set $\operatorname{ND}_{\ell_{2}}(X,\pi)$. Note that this set
is closed as $\operatorname{ND}_{\ell_{2}}(X,\pi)$ is compact (5.2). The Hahn-
Banach theorem ensures the existence of a half-space $H$ which is disjoint
from $\operatorname{ND}_{\ell_{2}}(X,\pi)$. 5.5 then provides a contradiction.

Since the origin belongs to the convex hull generated by the set
$\operatorname{ND}_{\ell_{2}}(X,\pi)$, Gordan’s theorem provides the
interpretation in terms of intersections. ∎

A natural question is to ask how optimal is the restriction on
$\operatorname{ND}_{\ell_{2}}(X)$ in 5.6. More precisely, we have the
following question:

######  Question 5.7.

Given a closed set $N\subseteq{\mathbb{S}}^{d-1}$ such that the origin of
${\mathbb{R}}^{d}$ belongs to its convex hull, does there exist a topological
dynamical system $(X,T,{\mathbb{Z}}^{d})$ such that
$\operatorname{ND}_{\ell_{2}}(X)=N$?

Of course, the former question also makes sense when restricted to some class
of dynamical systems with specific properties (like transitivity, minimality,
subshift of finite type like in [30], etc.).

It is already a challenging question for simple cases: for instance, we do not
know if there is a ${\mathbb{Z}}^{2}$-action where
$\operatorname{ND}_{\ell_{2}}(X)$ consists of three vectors, two of them
opposites and a third one orthogonal to the others.

Previous results in the literature are not enough to answer the question but
provide a partial answer. Recall that Boyle and Lind [7] and Hochman [17]
proved that any closed set of directions can be the set of nonexpansive
directions, which we denote by $\operatorname{NE}(X)$, of a topological
dynamical system $(X,T,{\mathbb{Z}}^{2})$. For instance, for any vector $u$
determining an (orthogonal) direction in $\operatorname{NE}(X)$, either $u$ or
$-u$ determines a nonexpansive $\ell_{2}$ horoball of this action. In
particular, the set $\operatorname{NE}(X)$ can be reduced to a point (even
defining a line with irrational slope). Actually, let
$\pi\colon{\mathbb{S}}^{1}\to{\mathbb{R}}{\textbf{\rm P}}^{1}$ denote the
covering onto the projective space, where the vector $u$ is mapped to the line
orthogonal to $u$, so that $u$ and its opposite $-u$ are identified. Under
this map one has that
$\operatorname{ND}_{\ell_{2}}(X)\subseteq\pi^{-1}(\operatorname{NE}(X))$.
Hence the local topological properties of $\operatorname{ND}_{\ell_{2}}(X)$
and $\operatorname{NE}(X)$ are the same. However, this inclusion can be strict
as Ledrappier’s example 4.2 shows: $\operatorname{ND}_{\ell_{2}}(X)$ consists
only of $3$ vectors and none of their opposites are in
$\operatorname{ND}_{\ell_{2}}(X)$.

One could ask the relative version of 5.7, that is: Given a topological
dynamical system $(Y,S,{\mathbb{Z}}^{d})$ and a closed set
$N\subseteq{\mathbb{S}}^{d-1}$ such that the origin of ${\mathbb{R}}^{d}$
belongs to its convex hull, does there exist a topological dynamical system
$(X,T,{\mathbb{Z}}^{d})$ which is an infinite extension of
$(Y,S,{\mathbb{Z}}^{d})$ and $\operatorname{ND}_{\ell_{2}}(X,\pi)=N$? This
question is actually equivalent to 5.7. To see this, assuming that there is a
positive answer to 5.7, given $N$ and $(Y,S,{\mathbb{Z}}^{d})$, there exists
$(X,T,{\mathbb{Z}}^{d})$ such that $\operatorname{ND}_{\ell_{2}}(X)=N$.
Consider $\pi\colon X\times Y\to Y$ the projection onto the second coordinate.
It is immediate to check that $\operatorname{ND}_{\ell_{2}}(X\times
Y,\pi)=\operatorname{ND}_{\ell_{2}}(X)=N$, which provides a positive answer to
the relative version of 5.7.

Recall that a horoball of any proper right-invariant distance on
${\mathbb{Z}}^{d}$ is included in a $\ell_{2}$ horoball (see the discussion
after 2.2). It follows that the conclusion of 5.6 is still true for any proper
right-invariant distance on ${\mathbb{Z}}^{d}$, namely the intersection of all
the non expansive horoballs is empty. This leads to ask

######  Question 5.8.

For which group/metrics, the intersection of all nonexpansive horoballs of any
action on an infinite space is empty?

The closeness of the set of nondeterministic vectors is important to get 5.6.
It is also natural to wonder if 5.2 is still true for metrics other than
$\ell_{2}$. More generally, is it always the case that the set of nonexpansive
horoballs corresponds to a closed subset of horoballs, for any distance in the
group? Before giving a partial answer to this question, let us precise the
topology we may consider on the set of horoballs. Since the topology on the
horofunctions is the compact open topology, a natural topology on the
horoballs is then given by the Hausdorff convergence on any compact set of the
group 333This means that $H_{n}\to H$ as $n\to\infty$ if and only if for all
compact set $K\subseteq G$ and $\epsilon>0$ the Hausdorff distance between
$\overline{H_{n}}\cap K$ and $\overline{H}\cap K$ is smaller than $\epsilon$
for all large enough $n$.. With this topology, the following simple example
shows that the set of nonexpansive horoballs is not always closed, even in
${\mathbb{Z}}^{2}$. As we shall see, the distance chosen on the group plays an
essential role.

######  Example 5.9 (A ${\mathbb{Z}}^{2}$-subshift with a nonclosed set of
nondeterministic $\ell_{1}$ horoballs).

Consider an infinite ${\mathbb{Z}}$-subshift $X$, i.e., a closed subset
$X\subseteq\\{0,1\\}^{\mathbb{Z}}$, invariant by the shift map $\sigma$.
Define the ${\mathbb{Z}}^{2}$ system $(X,T,{\mathbb{Z}}^{2})$ endowed with the
$\ell_{1}$-norm where $T_{(n,m)}=\sigma^{n-2m}$. One checks that the
$\ell_{1}$ horoball $\\{(x,y):-x<y<x\\}$ is nonexpansive as well as all its
translates along the half line $y=x$, $y>0$. If the set of nonexpansive
horoballs for $\ell_{1}$ were closed, the horoball $\\{(x,y):y<x\\}$ would be
a nonexpansive horoball for this subshift. But, as the action is the identity
along the line $\\{2y=x\\}\cap{\mathbb{Z}}^{2}$, it is straightforward to
check that this is imposible

This example leads to the following question related to the geometry of
groups.

######  Question 5.10.

Do there exist conditions on the group, the metric and/or the system so that
the set of nonexpansive horoballs of any action is a closed set? Are there
other interesting topologies on the set of horoballs for which the set of
nonexpansive horoballs is closed?

##  6\. Distality in Cantor dynamics

In this section we illustrate further applications of Robinson Crusoe theorem
(3.1) in the context of zero dimensional dynamics, i.e., when the phase space
$X$ is a zero dimensional space.

Recall that for a topological dynamical system $(X,T,G)$ a point $x\in X$ is
almost periodic if for every neighborhood $U$ of $x$ the set of return times
$N(x,U)=\\{g\in G:T_{g}(x)\in U\\}$ is syndetic (i.e., there exists a compact
set $K\subseteq G$ such that $KN(x,U)=G$). Equivalently, $x$ has a minimal
orbit closure, i.e., $(\overline{Gx},G)$ is minimal. The topological dynamical
system $(X,T,G)$ is pointwise almost periodic if every $x\in X$ is almost
periodic.

The following result was obtained in [4] (stated for finitely generated
groups). Here we give a proof for any countable group whose horofunctions
satisfy the conditions of 2.1.

######  Proposition 6.1.

Let $(X,T,G)$ be a dynamical system with $X$ a zero dimensional space and $G$
a countable group. Assume that the group admits a right-invariant and proper
metric so that each horoball $H$ contains arbitrarily large balls. Then the
following statements are equivalent,

  1. (1)

the action of $G$ is pointwise almost periodic;

  2. (2)

the relation $R=\\{(x,y)\in X\times X:y\in\overline{Gx}\\}$ is closed.

We notice that the conclusion of 6.1 is still true when the acting group $G$
is finite.

###### Proof.

The implication (2) $\Rightarrow$ (1) is valid for any topological $G$-action.
By contradiction, assume there exists $x\in X$ that is not almost periodic.
Then, there exists an almost periodic point $y\in\overline{Gx}$ such that
$x\notin\overline{Gy}$. Let $(g_{n})_{n\in{\mathbb{N}}}\subseteq G$ be a
sequence such that $T_{g_{n}}(x)$ converges to $y$ as $n$ goes to infinity. It
follows that $(T_{g_{n}}(x),x)\in R$ converges to $(y,x)\in R$, a
contradiction.

We also prove (1) $\Rightarrow$ (2) by contradiction. Assume that $R$ is not
closed. Then there exists a sequence $((x_{n},y_{n}))_{n\in{\mathbb{N}}}$ in
$R$ converging to $(x,y)\notin R$. Let $U$ be a clopen subset such that
$\overline{Gx}\subseteq U$ and $y\notin U$. Let $O$ be the set
$\\{y^{\prime}\in\overline{Gx}^{c}:\overline{Gy^{\prime}}\cap
U^{c}\neq\emptyset\\}$. One checks that $O$ is a $G$-invariant open set, which
is not closed since $x_{n}$ belongs to $O$ for large enough $n$. Moreover, $U$
is also a neighborhood of $\partial O$. It follows from Theorem 3.1 that there
exist $y^{\prime}\in O$ and a horoball $H$ such that $T_{g}(y^{\prime})\in U$
for every $g\in H$. Since each horoball contains arbitrarily large balls, the
set of return times $N(y",U^{c})$ of some point $y"\in U^{c}$ in the orbit of
$y^{\prime}$, cannot be syndetic. This is a contradiction. ∎

It follows from 2.1 that any finitely generated group fulfills the assumptions
of 6.1. It would be interesting to exhibit other groups or families of groups
that satisfy the assumptions of 6.1. For now, we do not rule out that this
family consists only of finitely generated groups.

Let $\pi\colon(X,T,G)\to(Y,S,G)$ be a factor map between the topological
dynamical systems $(X,T,G)$ and $(Y,S,G)$. We say that $\pi$ is equicontinuous
(or that $X$ is an equicontinuous extension of $Y$) if for any $\epsilon>0$
there exists $\delta>0$ such that ${\rm dist}(x,y)<\delta$ and $\pi(x)=\pi(y)$
implies that ${\rm dist}(T_{g}(x),T_{g}(y))<\epsilon$ for all $g\in G$. The
factor map $\pi$ is distal (or $X$ is a distal extension of $Y$) if
$\displaystyle\inf_{g\in G}{\rm dist}(T_{g}(x),T_{g}(y))>0$ whenever
$\pi(x)=\pi(y)$ and $x\neq y$. If the factor $\pi$ is trivial one says that
$(X,T,G)$ is equicontinous, or distal, respectively.

Two points $x,y\in X$ are $\pi$-regionally proximal if for any $\epsilon>0$
there exist points $x^{\prime},y^{\prime}\in X$ and $g\in G$ such that
$\pi(x^{\prime})=\pi(y^{\prime})$, ${\rm dist}(x,x^{\prime})<\epsilon$, ${\rm
dist}(y,y^{\prime})<\epsilon$ and ${\rm
dist}(T_{g}x^{\prime},T_{g}y^{\prime})<\epsilon$. We let ${\bf RP}(\pi)$
denote the set of $\pi$-regionally proximal points. It is well known (see for
instance [3, Chapter 7, Theorem 2]) that a factor map is equicontinuous if and
only if ${\bf RP}(\pi)=\triangle_{X}$ and that if a factor map is
equicontinuous, then it is a distal factor.

######  Corollary 6.2.

Let $\pi\colon(X,T,G)\to(Y,S,G)$ be a factor map, where $G$ is a countable
group, $X$ is zero dimensional and $(Y,S,G)$ is a pointwise almost periodic
system. Assume that the group admits a right-invariant and proper metric so
that each horoball $H$ contains arbitrarily large balls. Then, the factor
$\pi$ is distal if and only if it is equicontinuous.

###### Proof.

We show the nontrivial implication. Assume that $\pi$ is distal and consider
the zero dimensional system $(R_{\pi},T^{(2)},G)$, where $R_{\pi}=\\{(x,y)\in
X\times X:\pi(x)=\pi(y)\\}$ and $T^{(2)}$ is the diagonal action
$T^{(2)}_{g}(x,y)=(T_{g}(x),T_{g}(y))$. We claim that the distality of $\pi$
implies that $(R_{\pi},G)$ is pointwise almost periodic. To show this, we need
to use the machinery of the enveloping semigroup (to avoid details about this
theory, we refer to the book [3, Chapters 3 and 6]). Let $E(X,T,G)$ and
$E(Y,S,G)$ denote the enveloping semigroup of the systems $(X,T,G)$ and
$(Y,S,G)$ respectively. Since $\pi(x)$ is an almost periodic point, we can
find $u\in E(Y,S,G)$ a minimal idempotent such that $u\pi(x)=\pi(x)$. We can
take a minimal idempotent $v\in E(X,T,G)$ such that $\pi_{\ast}(v)=u$, where
$\pi_{\ast}\colon E(X,T,G)\to E(Y,S,G)$ is the natural induced semigroup
homomorphism (see [3, Chapter 3, Theorem 7]). Then, $v(x,y)=(vx,vy)$ and,
since $u\pi(x)=\pi(x)$, we obtain $\pi(vx)=\pi(x)=\pi(y)=\pi(vy)$. Noting that
$(x,vx)$ and $(y,vy)$ are proximal pairs (i.e., $\inf_{g\in G}{\rm
dist}(x,vx)=\inf_{g\in G}{\rm dist}(y,vy)=0$), we obtain that $x=vx$ and
$y=vy$. Hence, $(x,y)=v(x,y)$ is a minimal point.

By 6.1, the orbit relation $R=\\{((x_{1},y_{1}),(x_{2},y_{2}))\in
R_{\pi}\times R_{\pi}:(x_{2},y_{2})\in\overline{G(x_{1},y_{1})}\\}$ is closed.
From this, it is classical to deduce that ${\bf RP}(\pi)=\triangle_{X}$, we
provide the argument for completeness. Let $(x,y)\in{\bf RP}(\pi)$ and let
$(x_{n},y_{n})\in R_{\pi}$ and $g_{n}\in G$ such that $(x_{n},y_{n})\to(x,y)$
and $(T_{g_{n}}x_{n},T_{g_{n}}y_{n})\to(z,z)$, for $z\in X$ and a sequence
$(g_{n})$ in $G$. Since $((T_{g_{n}}x_{n},T_{g_{n}}y_{n}),(x_{n},y_{n}))\in
R$, and this relation is closed, we obtain that $((z,z),(x,y))\in R$. This
implies that $x=y$. We conclude that $\pi$ is an equicontinuous extension.

∎

When $(Y,S,G)$ is the trivial system, we obtain the following corollary.

######  Corollary 6.3.

Let $(X,T,G)$ be a topological dynamical where $X$ is zero dimensional and $G$
is countable. Assume that the group $G$ admits a proper and right-invariant
metric so that each horoball contains arbitrarily large balls. Then, the
action of $G$ is distal if and only if it is equicontinuous.

We remark that 6.3 was obtained in [4] under the assumption that $G$ is
finitely generated. Moreover, 6.3 does not hold for all countable groups. In
[22] (page 266), the authors constructed examples of (non finitely generated)
groups that admit distal but non equicontinuous actions on a Cantor space. We
thank Benjy Weiss and Xiangdong Ye for pointing us to this example and
reference.

The next corollary is Theorem 1.3 in [23]. Unlike [23], we derive it as a
consequence of 6.3.

For a group $G$, and a subgroup $S\leqslant G$, we let ${\rm Cent}_{G}(S)$
denote the centralizer of $S$ in $G$, i.e., the set of elements $g\in G$ such
that $gs=sg$ for all $s\in S$. It is a group under composition. For a
topological dynamical system $(X,T,G)$, let ${\rm Homeo}(X)$ denote the group
of all the self-homeomorphisms of $X$ and $\langle T\rangle$ the group
generated by the elements $T_{g}\in{\rm Homeo}(X)$, $g\in G$.

######  Corollary 6.4.

[23] Let $(X,T,G)$ be a $G$-subshift for a countable group $G$. Let
$N\leqslant{\rm Cent}_{{\rm Homeo}(X)}(\langle T\rangle)$ be a finitely
generated subgroup. If each $N$-orbit in $X$ is finite, then the group $N$ is
finite.

In particular, when $N$ is reduced to shift maps, there is no infinite
subshift for a finitely generated group $G$ where all the shift-orbits are
finite.

###### Proof.

Since the shift action is expansive and $X$ is zero dimensional, it is well
known that the centralizer group ${\rm Cent}_{{\rm Homeo}(X)}(\langle
T\rangle)$ is discrete in ${\rm Homeo}(X)$ for the uniform topology. If each
$N$-orbit is finite, its action is distal. It follows from Corollary 6.3 that
the action is equicontinuous, hence $N$ is a compact subgroup of ${\rm
Homeo}(X)$. As a discrete group, it is finite. ∎

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