# Strongly isomorphic symbolic extensions for expansive topological flows

Yonatan Gutman  Institute of Mathematics, Polish Academy of Sciences, ul.
Śniadeckich 8, 00-656 Warszawa, Poland<EMAIL_ADDRESS>and  Ruxi Shi
Sorbonne Université, LPSM, 75005 Paris, France<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove that finite-dimensional topological flows without
fixed points and having a countable number of periodic orbits, have the small
flow boundary property. This enables us to answer positively a question of
Bowen and Walters from 1972: Any expansive topological flow has a strongly
isomorphic symbolic flow extension, i.e. an extension by a suspension flow
over a subshift. Previously Burguet had shown this is true if the flow is
assumed to be $C^{2}$-smooth.

Y.G. was partially supported by the National Science Centre (Poland) grant
2020/39/B/ST1/02329.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Notation
    2. 2.2 Discrete dynamical systems and topological flows
    3. 2.3 Cross-sections
    4. 2.4 Flow boundaries and interiors
    5. 2.5 Good cross-sections exist
    6. 2.6 Calculating with flow interiors and boundaries
    7. 2.7 The small flow boundary property
    8. 2.8 Suspension flows and symbolic extensions
    9. 2.9 Some facts from dimension theory
  3. 3 Establishing the small flow boundary property
    1. 3.1 Local homeomorphisms between cross-sections
    2. 3.2 Return time sets and cross-section names
    3. 3.3 Statement of the Main Theorem
    4. 3.4 General position
    5. 3.5 Key Lemma
    6. 3.6 Proof of the Main Theorem
  4. 4 Expansive flows have strongly isomorphic symbolic extensions
    1. 4.1 Background
    2. 4.2 Bowen and Walters’ construction
    3. 4.3 Bowen and Walters’ question answered
  5. 5 Appendix
    1. 5.1 Existence of a complete family

##  1\. Introduction

Symbolic coding of dynamical systems $(X,T)$ in the form of measurable factor
maps into shift spaces over finite alphabets
$(X,T)\rightarrow(Y,\sigma)\subset(\\{1,2,\ldots,a\\}^{\mathbb{Z}},\sigma)$
has played a prominent role in the theory of dynamical systems since its
inception ([Had98, Mor21, MT88]). Requiring the factor maps to be continuous
is usually impossible. It is however meaningful to look for a symbolic system
$(Y,\sigma)\subset(\\{1,2,\ldots,a\\}^{\mathbb{Z}},\sigma)$ for which the
original system occurs as a continuous factor $(Y,\sigma)\rightarrow(X,T)$. In
other words this form of “digitization” corresponds to symbolic extensions.
However as $(Y,\sigma)$ is an extension of $(X,T)$ it is necessarily “more
complex” than $(X,T)$. Viewing $(Y,\sigma)$ as a model of $(X,T)$ one strives
to minimize this “complexity gap”. This may be formalized mathematically by
various conditions such as requiring the extension to be principal or strongly
isomorphic111See Definition 2.22. The associated theory for
$\mathbb{Z}$-systems is deep and extensive ([BD04, Dow05, Dow11]). Recently a
symbolic extension theory for $\mathbb{R}$-systems, i.e. topological flows has
been put forth ([Bur19]). The present paper is a further contribution in this
direction, specifically to the theory of symbolic extensions of expansive
topological flows. Our main tool is the small flow boundary property.

As a dynamical analog of Lebesgue covering dimension zero, the small boundary
property for a $\mathbb{Z}$-system $(X,T)$ was introduced by Shub and Weiss in
[SW91] who investigated the question under which conditions a given
$\mathbb{Z}$-system has factors with strictly lower entropy. Later it was
realized that the small boundary property has wider applicability. Notably
Lindenstrauss and Weiss [LW00] showed that a $\mathbb{Z}$-system which has the
small boundary property must have mean dimension zero. Moreover a
$\mathbb{Z}$-system with the small boundary property has a zero-dimensional
strongly isomorphic extension ([Bur19, p. 4338] based on [Dow05]). From
[Lin95] it follows that a finite-dimensional $\mathbb{Z}$-system without
periodic points has the small boundary property222In effect from [Lin95,
Theorem 3.3] it follows that an infinite $\mathbb{Z}$-system
($|X|~{}=~{}\infty)$ with a finite number of periodic points has the small
boundary property..

Burguet [Bur19] introduced the small flow boundary property for topological
flows as an analog to the small boundary property. He showed that flows with
the small flow boundary property admit strongly isomorphic zero-dimensional
extensions and gave necessary and sufficient conditions for the existence of
symbolic extensions333A topological flow is said to admit a symbolic
extension, respectively a zero-dimensional extension, if it has an extension
by a suspension flow over a subshift, respectively a zero dimensional system,
with a positive continuous roof function. for such flows in terms of the
existence of superenvelopes ([Bur19, Theorem 3.6]). This can be seen as a
certain generalization of the Boyle-Downarowicz symbolic extension entropy
theorem ([BD04]). Burguet [Bur19] showed that a $C^{2}$-flow without fixed
points444Flows without fixed points are known as _regular flows_ but we will
not use this terminology in this paper. and such that for any $\tau>0$, the
number of periodic orbits of period less than $\tau$ is finite has the small
flow boundary property. In our main theorem we manage to remove the smoothness
assumption:

######  Theorem A.

Let $X$ be a compact finite-dimensional space. Let $\Phi$ be a topological
flow on $X$ without fixed points, having a countable number of periodic
orbits. Then $(X,\Phi)$ has the small flow boundary property.

Expansive $\mathbb{Z}$-systems were introduced as early as 1950 by [Utz50]. In
[Red68] and [KR69] it was shown that expansive $\mathbb{Z}$-systems admit
symbolic extensions555From the work of Boyle and Downarowicz [BD04] it follows
that an expansive $\mathbb{Z}$-system admits a symbolic extension of the same
entropy.. The notion of expansiveness for flows was introduced by Bowen and
Walters [BW72]. Bowen and Walters [BW72] proved that expansiveness is
invariant under topological conjugacy and that the topological entropy of an
expansive flow is finite. In addition they constructed a symbolic extension
for expansive flows. They asked whether this symbolic extension preserves
entropy. More precisely, they made use of closed cross-sections to build a
symbolic extension and wondered if one could choose carefully these closed
cross-sections so that the associated symbolic extension has the same
topological entropy as the original system. Burguet [Bur19] gave a positive
answer to this question for $C^{2}$-expansive flows. In this paper, we give an
affirmative answer for all expansive flows.

######  Theorem B.

Let $(X,\Phi)$ be an expansive flow. Then it has a strongly isomorphic
symbolic extension.

### Structure of the paper

In Section 2, we recall basic notions related to discrete dynamical systems
and topological flows. In Section 3, we prove that finite-dimensional
topological flows without fixed points, having a countable number of periodic
orbits, satisfy the small flow boundary property (Theorem A). In Section 4, we
recall the definition of expansive flows and Bowen-Walters construction of
symbolic extensions for expansive flows, and show that any expansive
topological flow has a strongly isomorphic symbolic extension (Theorem 4.12).
This answers an open question of Bowen and Walters [BW72]. In the Appendix
(Section 5), we review the construction of a complete family of cross-sections
for a topological flow without fixed points, following Bowen and Walters.

### Acknowledgements

We are grateful to David Burguet who told us about the main problem of the
paper and conveyed to us his strong conviction in the feasibility of a
positive solution (see also [Bur19, Remark 2.3]).

##  2\. Preliminaries

###  2.1. Notation

Let $(X,d)$ be a metric space. Let $x\in S\subset X$. Denote by
$B_{S}(x,\epsilon)=\\{y\in S|\,d(x,y)<\epsilon\\}$ the open $\epsilon$-ball in
$S$ around $x\in S$. If $S$ is clear from the context, it may be omitted from
the notation. Let $C=\\{A_{1},\ldots A_{n}\\}$ be a set of sets $A_{i}\subset
X$, $i=1,\ldots,n$. We denote by $\bigcup C=\cup_{i=1}^{n}A_{i}\subset X$ and
$\bigcap C=\cap_{i=1}^{n}A_{i}\subset X$. Let $S,V\subset X$. Denote by
$\partial_{S}V$ and $\text{\rm Int}_{S}V$ respectively the boundary and
interior of $V$ w.r.t. the subspace topology induced by $S$. Let $S\subset X$
be a closed subset and $Q\subset S$ be a subset in $S$. Fix $\epsilon>0$. The
open, respectively closed $\epsilon$-tube around $Q$ is the set
$\Theta^{S}_{\epsilon}(Q):=\\{y\in S|\,d(y,\overline{Q})<\epsilon\\}$
respectively $\overline{\Theta}^{S}_{\epsilon}(Q):=\\{y\in
S|\,d(y,\overline{Q})\leq\epsilon\\}$. Let $O\subset S$ be an open subset in
$S$. The open, respectively closed internal $(-\epsilon)$-tube of $O$ inside
$S$ is the set $\Theta^{S}_{-\epsilon}(O):=\\{y\in O|\,d(y,S\setminus
O)>\epsilon\\}$ respectively $\overline{\Theta}^{S}_{-\epsilon}(O):=\\{y\in
O|\,d(y,S\setminus O)\geq\epsilon\\}$. If $S$ is clear from the context, it
may be omitted from the notation.

###  2.2. Discrete dynamical systems and topological flows

A pair $(X,T)$ is called a (discrete) dynamical system if $X$ is a compact
metric space and $T:X\to X$ is a homeomorphism. For $Y$, a second countable
metrizable space we denote by $\dim(A)$ the Lebesgue covering dimension of
$A$.666For second countable normal spaces (metrizable by the Urysohn
metrization theorem), the Lebesgue covering dimension equals the small
inductive dimension (see [Fed90, §6.2]). We use the convention $\dim(B)=-1$
iff $B=\emptyset$.

######  Definition 2.1.

A topological flow is a pair $(X,\Phi)$ where $X$ is a compact metrizable
space and $\Phi:X\times\mathbb{R}\to X$ is a continuous flow on $X$, that is,
the map $\Phi$ is continuous, $\Phi(\cdot,0)$ is the identity map on $X$ and
$\Phi(\Phi(x,t),s)=\Phi(x,t+s)$ for all $t,s\in\mathbb{R}$ and $x\in X$. For
$t\in\mathbb{R}$, we sometimes use the notation $\Phi_{t}(x)=\Phi(x,t)$ and
notice that $\Phi_{t}:X\rightarrow X$ is a homeomorphism. In addition for
$L\subset\mathbb{R}$, $S\subset X$ we denote
$\Phi_{L}(S)=\\{\Phi_{t}(x)\,|\,t\in L,\,x\in S\\}$. Throughout the text, we
fix a compatible metric $d$ on $X$. Given a set $\emptyset\not=A\subset X$ and
$x\in X$, we define $d(x,A)=\inf_{y\in A}d(x,y)$, as well as
$d(x,\emptyset)=\infty$.

######  Definition 2.2.

A point $x\in X$ is called a fixed point if $\Phi_{t}(x)=x$ for all $t$. A
point $x\in X$ is a periodic point with period $\tau>0$ if $\Phi_{\tau}(x)=x$
and $\Phi_{t}(x)\not=x$ for any $0<t<\tau$. In the later case, the set
$\\{\Phi_{t}(x)\,|\,0\leq t\leq\tau\\}$ is called the periodic orbit
associated with $x$.

######  Definition 2.3.

The flow $(X,\Phi)$ is said to be aperiodic if it has no periodic orbits, that
is, the equation $\Phi_{t}(x)=x$ implies $t=0$.

###  2.3. Cross-sections

######  Definition 2.4.

A cross-section of injectivity time $\eta>0$ is a subset $S\subset X$ such
that the restriction of $\Phi$ on $S\times[-\eta,\eta]$ is one-to-one. The
cross-section is said to be global if there is $\xi>0$ such that
$\Phi(S\times[-\xi,\xi])=X$. The set $\Phi_{[-\eta,\eta]}(S)$ is called the
$\eta$-cylinder associated with $S$.

######  Remark 2.5.

In [Bur19] a closed cross-section $S$ such that the flow map
$\Phi:(x,t)\rightarrow\Phi_{t}(x)$ is a surjective local homeomorphism from
$S\times\mathbb{R}$ to $X$, is called a Poincaré cross-section and it is shown
this is strictly stronger than $S$ being a closed and global cross-section.

######  Definition 2.6.

Let $(X,\Phi)$ be a topological flow. A finite family
$\SS=\\{S_{i}\\}_{i=1}^{N}$ of disjoint closed cross-sections each of
injectivity time $\eta$ is said complete if
$\bigcup_{i=1}^{N}\Phi_{[-\frac{\eta}{2},\frac{\eta}{2}]}(S_{i})=X$. It
follows that $\mathcal{G}=\cup_{i=1}^{N}S_{i}$ is a closed global cross-
section.

A trivial observation is if $x\in S$ and $\Phi_{t}x\in S$ for some $t\not=0$,
where $S$ is a cross-section of injectivity time $\eta>0$, then $|t|>2\eta$ as
otherwise $\Phi_{-\frac{t}{2}}S\cap\Phi_{\frac{t}{2}}S=\emptyset$.

###  2.4. Flow boundaries and interiors

As we will see in the sequel, the cylinders associated with cross-sections
will play a fundamental role in the analysis of topological flows. The
following definition which originates in [BW72, Definition 3] is based upon
[Bur19, Lemma 2.1].

######  Definition 2.7.

Let $U$ be a set contained in a closed cross-section of injectivity time
$\eta$. The flow interior $\text{\rm Int}^{\Phi}(U)\subset U$ of $U$ is the
unique set obeying:

$\Phi_{(-\eta,\eta)}(\boldsymbol{{\rm Int}^{\Phi}(U)})={\rm
Int}(\Phi_{[-\eta,\eta]}(U))$

The flow boundary $\partial^{\Phi}U$ of $U$ is the unique set obeying:

$\partial\Phi_{[-\eta,\eta]}(U)=\Phi_{-\eta}(U)\sqcup\Phi_{\eta}(U)\sqcup\Phi_{(-\eta,\eta)}(\boldsymbol{\partial^{\Phi}U})$

One can show that for every $0<\gamma<\eta$:

$\text{\rm Int}^{\Phi}(U)=\text{Int}(\Phi_{[-\gamma,\gamma]}(U))\cap U,$
$\partial^{\Phi}U=\overline{U}\setminus\text{\rm Int}^{\Phi}(U).$

Note that $\partial^{\Phi}U$ is closed as it can be written as
$\partial^{\Phi}U=\overline{U}\setminus\text{Int}(\Phi_{[-\gamma,\gamma]}).$
for any $0<\gamma<\eta$. We remark that under certain circumstances the above
notions coincide with the classical notions of interior and boundary under the
induced topology on the cross-section. See Proposition 2.15.

###  2.5. Good cross-sections exist

Here are several important facts regarding cross-sections:

######  Theorem 2.8.

(Whitney) [Whi33, page 270] Let $(X,\Phi)$ be a topological flow without fixed
points, then for each $x\in X$ there is a closed cross-section $S$ such that
$x\in\text{\rm Int}^{\Phi}S$.

###### Proof.

(sketch) Fix $x\in X$. For $y\in X$, define

$\theta(y)=\int_{0}^{1}d(\Phi_{s}(y),x)ds,$

where $d$ is the metric on $X$ compatible with the topology. It is easy to see
that for fixed $y\in X$ the function $t\mapsto\theta(\Phi_{t}(y))$ has a
continuous derivative. We denote the derivative at $t=0$ by
$\theta^{\prime}(y)$. An easy calculation shows:

$\theta^{\prime}(y)=d(\Phi_{1}(y),x)-d(y,x).$

Assume w.l.o.g. that $\Phi_{1}(x)\not=x$. Thus
$\theta^{\prime}(x)=d(\Phi_{1}(x),x)>0$. Using the inverse function theorem we
find $\ell>0$ such that
$\theta(\Phi_{-\ell}(x))<\theta(x)<\theta(\Phi_{\ell}(x))$ and
$\theta^{\prime}(\Phi_{t}(x))>0$ for all $-2\ell\leq t\leq 2\ell$. Using the
continuity of $\Phi$, $\theta$ and $\theta^{\prime}$ we may find an open
neighborhood $U$ of $x$ such that for all $y\in\overline{U}$,
$\theta^{\prime}(\Phi_{t}(y))>0$ for all $-2\ell\leq t\leq 2\ell$ and:

$\theta(\Phi_{-\ell}(y))<\theta(x)<\theta(\Phi_{\ell}(y)).$

Thus for every $y\in\overline{U}$ there is a unique $-\ell<t_{y}<\ell$ such
that $\theta(\Phi_{t_{y}}(y))=\theta(x)$. It is not hard to show that the set
$S=\\{\Phi_{t_{y}}(y)\,|\,y\in\overline{U}\\}$ is a closed cross-section such
that $x\in S\cap U\subset\text{\rm Int}^{\Phi}S$. Indeed $\ell>0$ is clearly
an injectivity time for $S$. Moreover $U\subset\Phi_{[-\ell,\ell]}(S)$, as if
$t_{y}<0$ then $t_{y}+\ell>0$ and if $t_{y}>0$ then $t_{y}-\ell<0$. ∎

Based on the previous theorem it is possible to prove:

######  Theorem 2.9.

(Bowen & Walters) A topological flow without fixed points admits a complete
family of (closed) cross-sections.

###### Proof.

See the proof of Lemma 5.1 where a stronger result is proven. ∎

######  Lemma 2.10.

Let $(X,\Phi)$ be a topological flow with $\dim(X)=n\geq 1$. Let $S\subset X$
be a closed cross-section. Then $\dim(S)\leq n-1$. If in addition $S$ is a
global cross-section then $\dim(S)=n-1$.

###### Proof.

As $S\times[-\eta,\eta]$ is homeomorphic to a subset of $X$, one has
$\dim(S\times[-\eta,\eta])\leq n$. By a theorem of Hurewicz in [Hur35], as $S$
is compact, $\dim(S)\leq n-1$. Now assume in addition that $S$ is a global
cross-section. Let $\xi>0$ such that the natural continuous map
$f:S\times[-\xi,\xi]\rightarrow X$ given by $(x,t)\mapsto\Phi(x,t)$ is
surjective. Note that $f$ is a closed map between two metric separable spaces
and for every $x\in X$, $f^{-1}(x)$ is countable. Thus by [Eng95, Theorem
1.12.4]

$\dim(X)\leq\dim(S\times[-\xi,\xi])+\sup_{x\in
X}\dim(f^{-1}(x))\leq\dim(S)+1+0,$

where we used the product theorem for non-empty metric separable spaces $A$
and $B$ $\dim(A\times B)\leq\dim(A)+\dim(B)$ ([Eng95, Theorem 1.5.16]). Thus
$\dim(S)\geq n-1$ as desired. ∎

###  2.6. Calculating with flow interiors and boundaries

Let $S,V\subset X$. Recall that $\partial_{S}V$ and $\text{\rm Int}_{S}V$
denote respectively the boundary and interior of $V$ w.r.t. the subspace
topology induced by $S$.

######  Lemma 2.11 ([Bur19], Lemma 2.2).

Let $S$ be a set contained in a closed cross-section. Let $U\subset S$. Then

$\partial_{S}U\subset\partial^{\Phi}U\subset\partial_{S}U\cup\partial^{\Phi}S.$

######  Example 2.12.

Consider a minimal rotation on the torus $(\mathbb{T}^{2},\Phi)$. Identify
$\mathbb{T}^{2}=[0,1]^{2}$ in the usual way. The closed set
$S=[A,B]\times\\{0\\}$, where $0<A<B<1$ is a global cross-section. Let $A<C<B$
and $U=[C,B]\times\\{0\\}$ be a strict subset of $S$. Then
$\partial_{S}U=\\{C\\}\times\\{0\\}$,
$\partial^{\Phi}U=\\{C,B\\}\times\\{0\\}$ and
$\partial^{\Phi}S=\\{A,B\\}\times\\{0\\}$. This is an example for which the
inclusion relations for sets in Lemma 2.11 are strict.

######  Lemma 2.13.

Let $S$ be a set contained in a closed cross-section. Let $U\subset S$. Then
$\partial^{\Phi}U=\partial_{S}U\cup(\overline{U}\cap\partial^{\Phi}S)$.

###### Proof.

By Lemma 2.11 and Definition 2.7 it follows on the one hand that
$\partial^{\Phi}U\subset(\partial_{S}U\cup\partial^{\Phi}S)\cap\overline{U}=\partial_{S}U\cup(\overline{U}\cap\partial^{\Phi}S)$.
On the other hand, since
$\overline{U}\cap\partial^{\Phi}S=\overline{U}\cap(\overline{S}\setminus\text{\rm
Int}^{\Phi}S)\subset\overline{U}\setminus\text{\rm
Int}^{\Phi}U=\partial^{\Phi}U$, we obtain by Lemma 2.11 that
$\partial_{S}U\cup(\overline{U}\cap\partial^{\Phi}S)\subset\partial^{\Phi}U$.
∎

######  Lemma 2.14.

Let $(X,\Phi)$ be a topological flow. Let $U$ be a subset of a closed cross-
section. Then $\text{\rm Int}^{\Phi}U=U$ implies that $U$ is relatively open
in $S$. If $U\cap\partial^{\Phi}S=\emptyset$, then the converse holds, i.e.
$U$ relatively open in $S$ implies $\text{\rm Int}^{\Phi}U=U$.

###### Proof.

Assume that $U=\text{\rm Int}^{\Phi}U=$. Fix $0<\gamma<\eta$. From the
definition of cross-section it is clear that $\Phi_{[-\gamma,\gamma]}(U)\cap
S\subset U$. Therefore $U=\text{\rm
Int}^{\Phi}U=\text{Int}(\Phi_{[-\gamma,\gamma]}(U))\cap
U=\text{Int}(\Phi_{[-\gamma,\gamma]}(U))\cap S$, implying that $U$ is
relatively open in $S$. Now assume that $U$ is (relatively) open in $S$, i.e.
$U\cap\partial_{S}U=\emptyset$. As it follows from Definition 2.7 that
$\text{\rm Int}^{\Phi}U\subset U$ and $\text{\rm
Int}^{\Phi}U\cup\partial^{\Phi}U=\overline{U}$, it is enough to prove
$\partial^{\Phi}U\cap U=\emptyset$. By Lemma 2.13, $\partial^{\Phi}U\cap
U=(\partial_{S}U\cap U)\cup(U\cap\partial^{\Phi}S)$. This expression equals
the empty set as by assumption $U\cap\partial^{\Phi}S=\emptyset$. ∎

Given a cross-section $S$ and $x\in\text{\rm Int}^{\Phi}S$, we clearly have
$\delta:=d(x,\partial^{\Phi}S)>0$ as $\partial^{\Phi}S$ is closed. Therefore
it is easy to see that $U=B(x,\delta/2)\cap S$ is a relatively open set in $S$
with $x\in U\subset\text{\rm Int}^{\Phi}S$ and
$\overline{U}\cap\partial^{\Phi}S=\emptyset$. This explains the usefulness of
the criterion in the following proposition. Indeed this proposition is key for
calculating flow boundaries and interiors in the theory we develop below.

######  Proposition 2.15.

Let $(X,\Phi)$ be a topological flow. Let $U$ be a subset of a closed cross-
section $S$ such that $\overline{U}\cap\partial^{\Phi}S=\emptyset$. Then:

$\partial^{\Phi}U=\partial_{S}U\,\,\mathrm{and}\,\,\text{\rm
Int}^{\Phi}U=\text{\rm Int}_{S}U.$

###### Proof.

By Lemma 2.13, $\partial^{\Phi}U=\partial_{S}U$. Thus $\text{\rm
Int}_{S}U=\overline{U}\setminus\partial_{S}U=\overline{U}\setminus\partial^{\Phi}U=\text{\rm
Int}^{\Phi}U$. ∎

######  Remark 2.16.

A case where the previous remark can be used is when $S$ is a cross-section,
$U,V\subset S$ with $\text{\rm Int}^{\Phi}U=U$ and $\text{\rm Int}^{\Phi}V=V$
and $\partial^{\Phi}U\subset V$. Indeed

$\overline{U}=\text{\rm Int}^{\Phi}U\cup\partial^{\Phi}U\subset\text{\rm
Int}^{\Phi}U\cup V=\text{\rm Int}^{\Phi}U\cup\text{\rm
Int}^{\Phi}V\subset\text{\rm Int}^{\Phi}S$

Thus $\overline{U}\cap\partial^{\Phi}S=\emptyset$.

###  2.7. The small flow boundary property

First, we recall the small boundary property for discrete dynamical system.

######  Definition 2.17.

Let $(X,T)$ a discrete dynnamical system. A subset $A\subset X$ has a small
boundary if $\mu(\partial A)=0$ for every $T$-invariant measure $\mu$. The
system $(X,T)$ is said to have the small boundary property if there is a basis
for the topology of $X$ consisting of open sets with small boundary.

Let $(X,\Phi)$ be a topological flow.

######  Definition 2.18.

A Borel subset of $X$ is called a null set if it has zero measure w.r.t. any
$\Phi$-invariant Borel probability measure. A Borel subset is said to be a
full set when its complement is a null set.

######  Definition 2.19.

A closed cross-section $S$ of injectivity time $\eta$ has a small flow
boundary if $\Phi_{[-\eta,\eta]}(\partial^{\Phi}(S))$ is a null set.

For a cross-section $A\subset X$, we define the counting orbit capacity of $A$
by

$\operatorname{\text{\rm
ocap}_{\sharp}}(A):=\lim_{T\to\infty}\frac{1}{T}\sup_{x\in X}\sharp\\{0\leq
t<T:\Phi_{t}(x)\in A\\}.$

The limit exists and is finite as $\sup_{x\in X}\sharp\\{0\leq
t<T:\Phi_{t}(x)\in A\\}$ is subadditive.

######  Lemma 2.20 ([Bur19], Lemma 2.10).

Let $(X,\Phi)$ be a topological flow. Suppose that $S$ is a closed cross-
section of injectivity time $\eta>0$. Then the following are equivalent.

  1. (1)

$S$ has a small flow boundary;

  2. (2)

$\operatorname{\text{\rm ocap}_{\sharp}}(\partial^{\Phi}S)=0$.

######  Definition 2.21.

Let $(X,\Phi)$ be a topological flow without fixed points. The flow $(X,\Phi)$
is said to have the small flow boundary property if for any $x\in X$ and any
closed cross-section $S^{\prime}$ with $x\in\text{\rm
Int}^{\Phi}(S^{\prime})$, there exists a subset $S\subset S^{\prime}$ such
that $x\in\text{\rm Int}^{\Phi}(S)$ and $S$ has a small flow boundary.

###  2.8. Suspension flows and symbolic extensions

Let $(Z,\rho)$ be a compact metric space and $T:Z\to Z$ a homeomorphism. Let
$f:Z\to\mathbb{R}_{>0}$ be a continuous map. Let $F_{f}=\\{(x,t):0\leq t\leq
f(x),x\in Z\\}\subset Z\times\mathbb{R}$. Let $R_{f}$ be the closed
equivalence relation on $F_{f}$ induced by $\\{((x,f(x)),(Tx,0))|\,x\in
Z\\}\subset F_{f}\times F_{f}$.

The suspension flow of $T$ under $f$ is the flow $\Phi$ on the space

$Z_{f}:=F_{f}/R_{f}$

induced by the time translation $T_{t}$ on $Z\times\mathbb{R}$ defined by
$T_{t}(x,s)=(x,t+s)$. A suspension flow over a zero-dimensional
$\mathbb{Z}$-topological dynamical system is called a zero dimensional
suspension flow and a topological extension by a zero-dimensional suspension
flow is said to be a zero-dimensional extension. Similarly a suspension flow
over a symbolic $\mathbb{Z}$-topological dynamical system (a.k.a
$\mathbb{Z}$-subshift) is called a  symbolic suspension flow and a topological
extension by a symbolic suspension flow is said to be a symbolic extension.

######  Definition 2.22.

Let $(X,\Phi)$ and $(Y,\Psi)$ be two topological flows. Suppose that $\pi:Y\to
X$ is a topological extension from $(X,\Phi)$ to $(Y,\Psi)$. A topological
extension is said to be (see [Bur19, §2.3])

  * •

entropy preserving when it preserves topological entropy i.e.
$\operatorname{h_{top}}(X,\Phi)=\operatorname{h_{top}}(Y,\Psi)$.

  * •

principal when it preserves the entropy of invariant measures, i.e.
$\operatorname{h}(\mu)=\operatorname{h}(\pi\mu)$ for all $\Psi$-invariant
measures $\mu$,

  * •

isomorphic when the map induced by $\pi$ on the sets of invariant Borel
probability measures is bijective and $\pi:(Y,\Psi,\mu)\to(X,\Phi,\pi\mu)$ is
a measure theoretical isomorphism for all $\Psi$-invariant measures $\mu$,

  * •

strongly isomorphic when there is a full set $E$ of $X$ such that the
restriction of $\pi$ to $\pi^{-1}E$ is one-to-one.

######  Remark 2.23.

Clearly, strongly isomorphic $\Longrightarrow$ isomorphic $\Longrightarrow$
principal$\Longrightarrow$ entropy preserving. It is easy to give an example
of an extension which is entropy preserving but not principal: Let
$\pi:(X,T)\rightarrow(Y,S)$ such that
$\operatorname{h_{top}}(T)>\operatorname{h_{top}}(S)$. Then let $f:X\sqcup
Y\to Y$ by $f(x)=\pi(x)$ if $x\in X$ and $f(x)=x$ if $x\in Y$. An example of
an extension which is principal but not isomorphic is given by [BD19, Theorem
4.7]. As far as we know, the question whether there exists an example of an
extension which is isomorphic but not strongly isomorphic, is open. Note that
the $1$-suspensions over the examples above give the analogous examples for
flows. Indeed by [Bur19, p. 4328], there is an affine homeomorphism between
the simplices of invariant measures
$\Theta:\mathcal{M}(X,T)\rightarrow\mathcal{M}(S_{1}(X),\Phi)$ given by
$\mu\mapsto\mu\times\lambda$, where $\lambda$ is the Lebesgue measure on the
interval $[0,1]$.

###  2.9. Some facts from dimension theory

The following results in dimension theory for a separable metric spaces will
be used in the proof of our main result. Recall that an $F_{\sigma}$ set is a
countable union of closed sets.

######  Theorem 2.24.

([Eng95, Proposition 1.2.12]) Let $E$ be a zero-dimensional subset in a
separable metric space $M$. Then for every $x\in M$ and every open
neighborhood $U$ of $x$ there is an open set $U^{\prime}\subset U$ with $x\in
U^{\prime}$ and $\partial U^{\prime}\cap E=\emptyset$

######  Theorem 2.25.

([HW41, Chapter III, Theorem III 2]) Let $(B_{i})_{i\in\mathbb{N}}$ be a
countable collection of closed sets in a separable metric space satisfying
$\dim B_{i}\leq k$ for all $i$. Then $\dim\bigcup_{i}B_{i}\leq k$.

The following corollary is obvious.

######  Corollary 2.26.

Let $(B_{i})_{i\in\mathbb{N}}$ be a countable collection of $F_{\sigma}$ sets
in a separable metric space satisfying $\dim B_{i}\leq k$ for all $i$. Then
$\dim\bigcup_{i}B_{i}\leq k$.

######  Lemma 2.27.

Let $M$ be a separable metric space. Let $C$ be closed and $U$ open then
$U\cap C$ is $F_{\sigma}$.

###### Proof.

Note $M$ is second-countable and write $U$ as a countable union of closed
balls. ∎

The following theorem is well known.

######  Theorem 2.28.

([Eng95, Theorem 1.5.7]) Let $M$ be a separable metric space. If
$-1<\dim(M)<\infty$ there is a zero-dimensional subset $E$ of $M$ such that
$\dim(M\setminus E)=\dim(M)-1$.

We will need the following strengthening of Theorem 2.28.

######  Theorem 2.29.

Let $M$ be a separable metric space. If $-1<\dim(M)<\infty$ there is a zero-
dimensional $F_{\sigma}$ subset $E$ of $M$ such that $\dim(M\setminus
E)=\dim(M)-1$.

###### Proof.

Denote $n=\dim(M)$. If $n=0$, set $E=M$. If $n=1$, let
$\\{B_{i}\\}_{i=1}^{\infty}$ be a countable basis of $M$ so that
$\dim(\partial B_{i})=0$ for all $i$. Let $E=\bigcup_{i}\partial B_{i}$. By
Theorem 2.25 $E$ is a zero-dimensional $F_{\sigma}$ set. Note
$\\{B_{i}\cap(M\setminus E)\\}_{i=1}^{\infty}$ is a basis of $M\setminus E$
with $\partial(B_{i}\cap(M\setminus E))=\emptyset$. Thus $\dim(M\setminus
E)=0$ as desired. Assume the claim has been established for $n-1\geq 0$. Let
$\\{B_{i}\\}_{i=1}^{\infty}$ be a countable basis of $M$ so that
$\dim(\partial B_{i})\leq n-1$ for all $i$. Using the inductive assumption let
$F_{i}\subset\partial B_{i}$ be a zero-dimensional $F_{\sigma}$ set so that
$\dim(\partial B_{i}\setminus F_{i})\leq n-2$. Let $E=\cup_{i}F_{i}$. By
Theorem 2.25, $E$ is a zero-dimensional $F_{\sigma}$ set. Note
$\\{B_{i}\cap(M\setminus E)\\}_{i=1}^{\infty}$ is a basis of $M\setminus E$
with $\partial(B_{i}\cap(M\setminus E))\subset\partial B_{i}\setminus F_{i}$.
Thus $\dim(M\setminus E)\leq n-1$. By [HW41, Chapter III, 2 B],
$\dim(E\cup(M\setminus E))\leq 1+\dim(E)+\dim(M\setminus E)$ and thus
$\dim(M\setminus E)=n-1$. ∎

As an illustration of Theorem 2.29 consider $M=[0,1]^{2}$ and note that
$E=M\cap(\mathbb{Q}\times\mathbb{Q})$ is zero-dimensional and $M\setminus E$
is one-dimensional. Indeed it is easy to see that balls centered at rational
coordinates with a rational radius form a base with zero-dimensional boundary
in $M\setminus E$. As an illustration of Theorem 2.24 note that balls centered
at rational coordinates with a transcendental radius do not intersect $E$.

######  Proposition 2.30.

Let $M$ be a metric compact space and $E$ a zero-dimensional subset of $M$.
Let $C$ be a closed set and $U$ an open subset such that $C\subset U\subset
M$. Then there is an open set $U^{\prime}$ with $C\subset
U^{\prime}\subset\overline{U^{\prime}}\subset U$ and $\partial U^{\prime}\cap
E=\emptyset$.

###### Proof.

First fix an open set $W$ so $C\subset W\subset\overline{W}\subset U$. Using
Theorem 2.24, for each $x\in C$, choose an open set $x\in U_{x}\subset W$ with
$\partial U_{x}\cap E=\emptyset$. Let $U_{x_{1}},\ldots,U_{x_{n}}$ be a cover
of $C$. It is easy to see that $U^{\prime}=\bigcup_{i=1}^{n}U_{x_{i}}\subset
W$ has the required properties. ∎

##  3\. Establishing the small flow boundary property

###  3.1. Local homeomorphisms between cross-sections

Let $(X,\Phi)$ be a topological flow without fixed points. Suppose that
$\dim(X)=d+1$ for $d\geq 0$. By Lemma 5.1 there exists an $\eta>0$ and
$0<\alpha<\eta$ such that $(X,\Phi)$ has two complete families of cross-
sections $\SS=\\{S_{i}\\}_{i=1}^{N}$ of (closed disjoint) cross-sections of
injectivity time $\eta_{\SS}$ and
$\SS^{\prime}=\\{S_{i}^{\prime}\\}_{i=1}^{N}$ of injectivity time
$\eta_{\SS}^{\prime}$ such that for all $1\leq i\leq N$,

  * •

$S_{i}=\overline{S_{i}}\subset\text{\rm Int}^{\Phi}(S_{i}^{\prime})$;

  * •

$\eta=\eta_{\SS}=\eta_{\SS^{\prime}}$;

  * •

$\max\\{\operatorname{diam}(S_{i}^{\prime})\\}\leq\alpha$;

  * •

$\Phi_{[0,\alpha]}\mathcal{G}=\Phi_{[-\alpha,0]}\mathcal{G}=X$, where
$\mathcal{G}=\cup_{S\in\mathcal{S}}S$.

######  Definition 3.1.

Let $1\leq i,j\leq N$. Denote by $t_{i,j}$ the first positive hitting time
from $S_{i}$ to $S_{j}$, that is, for $x\in S_{i}$,

$t_{i,j}(x)=\min\\{t>0:\Phi_{t}(x)\in S_{j}\\},$

where if the argument set is empty we put $t_{i,j}(x)=\infty$. The functions
$t_{i,j}^{\prime}$ are defined similarly w.r.t. $S_{i}^{\prime}$ and
$S_{j}^{\prime}$. Note $t_{i,j}^{\prime}(x)\leq t_{i,j}(x)$ for all $x\in
S_{i}$. Define

$D_{i,j}=\\{x\in S_{i}:t_{i,j}(x)\leq\eta\\}$

and

$D_{i,j}^{\prime}=\\{x\in S_{i}^{\prime}:t_{i,j}^{\prime}(x)<\infty\\}.$

Note $D_{i,i}=\emptyset$ for all $i$. Let $T_{i,j}$ be the first positive
hitting map from $D_{i,j}$ to $S_{j}$, that is, for $x\in D_{i,j}$,

$T_{i,j}(x)=\Phi_{t_{i,j}(x)}(x),$

Similarly, we define $T_{i,j}^{\prime}$ from $D_{i,j}^{\prime}$ to
$S_{j}^{\prime}$.

######  Definition 3.2.

Define the first return time map
$t^{(1)}_{\mathcal{G}}(x)=t_{\mathcal{G}}~{}:~{}\mathcal{G}~{}=~{}\cup_{S\in\mathcal{S}}S\to\mathbb{R}_{+}$
by

$t^{(1)}_{\mathcal{G}}(x)=t_{\mathcal{G}}(x):=\min\\{t>0|\,\Phi_{t}(x)\in\mathcal{G}\\}.$

As $\SS=\\{S_{i}\\}_{i=1}^{N}$ is a family of closed disjoint cross-sections,
there is $\gamma>0$ so that $S=\bigcup_{i=1}^{N}S_{i}$ is a cross-section with
injectivity time $\gamma$. Thus the argument set in the definition above is
never empty.

Inductively for $i\geq 1$ define the $(i+1)$-th return time map
$t^{(i+1)}_{\mathcal{G}}(x)~{}:~{}\mathcal{G}~{}=~{}\cup_{S\in\mathcal{S}}S\to\mathbb{R}_{+}$
by

$t^{(i+1)}_{\mathcal{G}}(x)=t^{(i)}_{\mathcal{G}}(t_{\mathcal{G}}(x))+t_{\mathcal{G}}(x).$

Define $t^{(0)}_{\mathcal{G}}(x)=0$ and $t^{(-k)}_{\mathcal{G}}(x)$ for
$k\in\mathbb{N}$, similarly to the above.

######  Definition 3.3.

For $1\leq i,j\leq N$, define:

$F_{i,j}:=\\{x\in D_{i,j}|\,t_{i,j}(x)=t_{\mathcal{G}}(x)\\}.$

That is $F_{i,j}$ is the set of $x\in S_{i}$ such that the first member of
$\SS$ it hits after leaving $S_{i}$ is $S_{j}$.

######  Lemma 3.4.

If $x\in F_{i,j}$, then $t_{i,j}(x)>\gamma$.

###### Proof.

Note $t_{\mathcal{G}}$ is bounded from below by $2\gamma$. ∎

######  Lemma 3.5.

For all $1\leq i\leq N$, it holds that $S_{i}=\cup_{j=1}^{N}F_{i,j}$.

###### Proof.

Let $x\in S_{i}$. As $\Phi_{[-\alpha,0]}\mathcal{G}=X$, there exists some $j$,
$t\in[0,\alpha]$ and $y\in S_{j}$ so that $\Phi_{-t}y=x$, i.e.,
$\Phi_{t}x=y\in S_{j}$ . Thus $t_{i,j}(x)\leq\alpha<\eta$ and $x\in D_{i,j}$.
Let $K=\\{1\leq k\leq N|\,x\in D_{i,k}\\}$. Note $j\in K$. Clearly there is
$k\in K$ so that $t_{i,k}=\min_{s\in K}t_{i,s}$. Conclude $x\in F_{i,k}$. ∎

######  Lemma 3.6.

Let $1\leq i,j\leq N$.

  1. (1)

If $x\in S_{i}^{\prime}$, $y\in S_{j}^{\prime}$ and $\Phi_{r}(x)=y$ for some
$0\leq r\leq 2\eta$, then $t_{i,j}^{\prime}(x)=r$.

  2. (2)

If $x\in D_{i,j}$, then $t_{i,j}^{\prime}(x)=t_{i,j}(x)$.

###### Proof.

For (1) assume for a contradiction that $t_{i,j}^{\prime}(x)<r$. It follows
that $\Phi(y,r-t_{i,j}^{\prime}(x))\in S_{j}^{\prime}$ and
$|t_{i,j}^{\prime}(x)-r|\leq 2\eta$. Since $\Phi$ is injective on
$S_{j}^{\prime}\times[-\eta,\eta]$, this is a contradiction.

For (2) note that by definition $t_{i,j}(x)\leq\eta$ and
$\Phi_{t_{i,j}(x)}(x)\in S_{j}^{\prime}$, thus by (1),
$t_{i,j}^{\prime}(x)=t_{i,j}(x)$. ∎

######  Proposition 3.7.

Let $x\in D_{i,j}$. Then there is an open neighborhood $V$ of $x$ in
$S_{i}^{\prime}$ with $\overline{V}\subset\text{\rm Int}^{\Phi}S_{i}^{\prime}$
such that $t_{i,j}^{\prime}$ is continuous on $V$, $t_{i,j}^{\prime}(x)\leq
2\eta$ for all $x\in V$ and $T_{i,j}^{\prime}|_{V}:V\rightarrow
S_{j}^{\prime}$ is an open map and $T_{i,j}^{\prime}|_{V}:V\rightarrow
T_{i,j}^{\prime}(V)$ is a homeomorphism.

###### Proof.

Let $0<\epsilon<\eta$. Since $T_{i,j}(x)\in S_{j}\in\text{\rm
Int}^{\Phi}S_{j}^{\prime}$, there is a $\kappa>0$ such that the open ball
$B(T_{i,j}(x),\kappa)$ is contained in $\Phi_{(-\epsilon,\epsilon)}(\text{\rm
Int}^{\Phi}S_{j}^{\prime})$. Since $\Phi$ is continuous, there is an
$\delta>0$ such that for any $y\in X$ with $d(x,y)<\delta$, we have that
$\Phi(y,t_{i,j}(x))\in
B(T_{i,j}(x),\kappa)\subset\Phi_{(-\epsilon,\epsilon)}(\text{\rm
Int}^{\Phi}S_{j}^{\prime})$. Let $V_{\delta}=\\{y\in
S_{i}^{\prime}:d(x,y)<\delta\\}$ which is an open neighborhood of $x$ in
$S_{i}^{\prime}$. As $x\in S_{i}$, $d(x,\partial^{\Phi}S_{i}^{\prime})>0$,
thus taking $\delta$ small enough, one may assume w.l.o.g. that
$\overline{V_{\delta}}\subset\text{\rm Int}^{\Phi}S_{i}^{\prime}$. It follows
by Lemma 3.6 that

(1) $t_{i,j}(x)-\epsilon<t_{i,j}^{\prime}(y)<t_{i,j}(x)+\epsilon,$

whenever $y\in V_{\delta}$. In particular $t_{i,j}^{\prime}(y)<2\eta$ for all
$y\in V_{\delta}$ and $T_{i,j}^{\prime}(V)\subset\text{\rm
Int}^{\Phi}S_{j}^{\prime}$. We claim that after taking $\delta$ small enough,
the map $t_{i,j}^{\prime}$ is continuous on $V=V_{\delta}$. If not, then there
exists a sequence $\\{y_{n}\\}_{n\in\mathbb{N}}\subset V$ such that
$\lim\limits_{n\to\infty}y_{n}=x$ but
$t_{x}:=\lim\limits_{n\to\infty}t_{i,j}^{\prime}(y_{n})\not=t_{i,j}(x)$. As by
definition $0<t_{i,j}(x)\leq\eta$ and by choice $\epsilon<\eta$, one has that
$t_{x}\leq t_{i,j}(x)+\epsilon<2\eta$ and $\Phi(x,t_{x})\in S_{j}^{\prime}$.
It follows that $\Phi(T_{i,j}(x),t_{x}-t_{i,j}(x))\in S_{j}^{\prime}$ and
$|t_{x}-t_{i,j}(x)|<2\eta$. Since $\Phi$ is injective on
$S_{j}^{\prime}\times[-\eta,\eta]$, this is a contradiction. Let
$z=T_{i,j}^{\prime}(x^{\prime})\in\text{\rm Int}^{\Phi}S_{j}^{\prime}$ for
some $x^{\prime}\in V$. Let $W_{\rho}=\\{y\in S_{j}^{\prime}:d(z,y)<\rho\\}$
be an open neighborhood of $z$ in $S_{j}^{\prime}$. Using Equation (1) with
$y=x^{\prime}$, for $\rho>0$ small enough, there is $0<\xi<\eta$ such that
$\Phi_{-t_{i,j}(x)}(\Phi_{(-\rho,\rho)}W_{\rho})$ is an open set in
$\Phi_{(-\xi,\xi)}(V)$. Thus for each $w\in W_{\rho}$, there are unique $v\in
V$, $r\in(-\xi,\xi)$, such that $\Phi_{t_{i,j}(x)-r}(v)=w$. By Lemma 3.6, one
must have $t_{i,j}^{\prime}(v)=t_{i,j}(x)-r$ and $T_{i,j}^{\prime}(v)=w$. This
implies $W_{\rho}\subset T_{i,j}^{\prime}(V)$ which implies
$T_{i,j}^{\prime}|_{V}:V\rightarrow S_{j}^{\prime}$ is an open map. In
addition, as $t_{i,j}^{\prime}$ is continuous on $V$ so is
$T_{i,j}^{\prime}|_{V}$. Thus in order to establish that
$T_{i,j}^{\prime}|_{V}:V\rightarrow T_{i,j}^{\prime}(V)$ is a homeomorphism,
it is enough to show that $T_{i,j}^{\prime}|_{V}$ is injective. Indeed if it
holds for $z_{1},z_{2}\in V$, that
$T_{i,j}^{\prime}(z_{1})=T_{i,j}^{\prime}(z_{2})$, that is
$\Phi_{t_{i,j}^{\prime}(z_{1})}(z_{1})=\Phi_{t_{i,j}^{\prime}(z_{2})}(z_{2})$,
then $z_{1}=\Phi_{t_{i,j}^{\prime}(z_{2})-t_{i,j}^{\prime}(z_{1})}(z_{2})$
which implies $|t_{i,j}^{\prime}(z_{2})-t_{i,j}^{\prime}(z_{1})|>2\eta$,
contradicting Equation (1). Q.E.D. ∎

###  3.2. Return time sets and cross-section names

######  Proposition 3.8.

For every $1\leq i,j\leq N$ there is a finite collection of open sets
$\mathcal{C}_{i,j}$ in $S_{i}^{\prime}$ with the following properties:

  1. (1)

$F_{i,j}\subset\bigcup\mathcal{C}_{i,j}\subset\overline{\bigcup\mathcal{C}_{i,j}}\subset\text{\rm
Int}^{\Phi}S_{i}^{\prime}$.

  2. (2)

For all $V\in\mathcal{C}_{i,j}$, $t_{i,j}^{\prime}$ is continuous on $V$ and
$t_{i,j}^{\prime}(x)\leq 2\eta$ for all $x\in V$.

  3. (3)

For all $V\in\mathcal{C}_{i,j}$, $(T_{i,j}^{\prime})_{|V}:V\rightarrow
T_{i,j}^{\prime}(V)$ is a homeomorphism and $T_{i,j}^{\prime}(V)$ is open in
$S_{j}^{\prime}$.

###### Proof.

Fix $1\leq i,j\leq N$. Let $x\in\overline{F_{i,j}}$. Let
$\\{x_{q}\\}_{q=1}^{\infty}\subset F_{i,j}\subset D_{i,j}$ so that
$x_{q}\rightarrow_{q\rightarrow\infty}x$. As $t_{i,j}(x_{q})\leq\eta$, we may
assume w.l.o.g. $t_{i,j}(x_{q})$ converges to some $t\leq\eta$. Clearly
$\Phi_{t}(x)\in S_{j}$ and therefore $x\in D_{i,j}$. Conclude
$\overline{F_{i,j}}\subset D_{i,j}\subset S_{i}\subset\text{\rm
Int}^{\Phi}S^{\prime}_{i}$. Using Proposition 3.7 cover $\overline{F_{i,j}}$
by a finite collection of open sets $\mathcal{C}_{i,j}$ such that
$\bigcup\mathcal{C}_{i,j}\subset\overline{\bigcup\mathcal{C}_{i,j}}\subset\text{\rm
Int}^{\Phi}S_{i}^{\prime}$. with properties (2) and (3). ∎

For each $C\in\mathcal{C}_{i,j}$, we define $T_{C}:C\to S_{j}^{\prime}$ by
$x\mapsto T_{i,j}^{\prime}(x)$. For $n\in\mathbb{N}\cup\\{0\\}$,
$\textbf{i}=(i_{k})_{0\leq k\leq n}\in\\{1,2,\dots N\\}^{n+1}$ and
$C^{n}=(C_{0},C_{1},\dots,C_{n-1},C_{n})\in\mathcal{C}(\textbf{i}):=\prod_{k=0}^{n-1}\mathcal{C}_{i_{k},i_{k+1}}\times\\{\mathcal{C}_{i_{n},j}\\}_{1\leq
j\leq N}$, we define

$T_{C^{n}}^{k}=\begin{cases}T_{C_{k-1}}\circ\cdots\circ
T_{C_{0}}~{}&\text{for}~{}1\leq k\leq n;\\\
\text{Id}_{C_{0}}&\text{for}~{}k=0;\end{cases}$

on the set

$Z_{C^{n}}:=\\{x\in C_{0}:T_{C_{k-1}}\circ\cdots\circ T_{C_{0}}(x)\in
C_{k}~{}\text{for}~{}1\leq k\leq n\\}.$

Let $x\in Z_{C^{n}}$. The set $Z_{C^{n}}$ is called an $n$-cross-section name
of $x$. Note $x$ may have more than one $n$-cross-section name for a given
$n\in\mathbb{N}$.

Let $V\subset X$, not necessarily contained in the image of $T_{C^{n}}$.
Following standard notation, we denote

(2) $T_{C^{n}}^{-k}(V):=\\{x\in Z_{C^{n}}:T_{C^{n}}^{k}(x)\in
V\\}=(T_{C^{n}}^{k})^{-1}(V\cap T_{C^{n}}^{k}(Z_{C^{n}})).$

Let $n\in\mathbb{N}\cup\\{0\\}$. Denote

$\mathcal{C}(n)=\cup_{\textbf{i}\in\\{1,2,\dots
N\\}^{n+1}}\mathcal{C}(\textbf{i}).$

$\mathcal{C}_{i}=\cup_{1\leq j\leq N}\mathcal{C}_{i,j}.$

######  Definition 3.9.

We define the return time set:

$I_{C^{n}}=\\{0\leq k\leq n:i_{k}=i_{0}\\}.$

It follows that the image of $T_{C^{n}}^{k}$ is contained in
$S_{i_{0}}^{\prime}$ for $k\in I_{C^{n}}$. Note $0\in I_{C^{n}}$.

######  Remark 3.10.

Note that $\mathcal{C}(0)=\bigcup_{1\leq i,j\leq N}\mathcal{C}_{i,j}$. Note
that for $C^{0}=(C_{0})\in\mathcal{C}(0)$ it holds $I_{C^{0}}=\\{0\\}$,
$Z_{C^{0}}=C_{0}$ and $T_{C^{0}}^{0}=\text{Id}_{C_{0}}$.

######  Lemma 3.11.

Let $C^{n}\in\mathcal{C}(n)$ and $1\leq k\leq n$. Then the following holds:

  1. (1)

$Z_{C^{n}}$ and $T_{C^{n}}^{k}(Z_{C^{n}})$ are open; $T_{C^{n}}^{k}$ is a
homeomorphism on $Z_{C^{n}}$.

  2. (2)

Let $V$ be $F_{\sigma}$. Then $T_{C^{n}}^{-k}(V)$ is $F_{\sigma}$.

###### Proof.

The first claim is best understood by considering the first cases. Indeed note
$T_{C^{n}}^{1}=T_{C_{0}}:C_{0}\rightarrow T_{i_{0},i_{1}}^{\prime}(C_{0})$ and
$T_{C^{n}}^{2}:T_{C_{0}}^{-1}(T_{C_{0}}(C_{0})\cap C_{1})\rightarrow
T_{C_{1}}(T_{C_{0}}(C_{0})\cap C_{1})$, both have a domain which is an an open
set in $S_{i_{0}}^{\prime}$ and a range which is an open set in
$S_{i_{1}}^{\prime}$ and $S_{i_{2}}^{\prime}$ respectively by Proposition 3.7.
Now assume we have proven that the domain of $T_{C^{n}}^{k}$ is an open set
$D_{k}$ in $S_{i_{0}}^{\prime}$ and that the range of $T_{C^{n}}^{k}$ is an
open set $R_{k}$ in $S_{i_{k}}^{\prime}$. It is easy to see that the domain of
$T_{C^{n}}^{k+1}$ is the open set in $D_{k}$ and therefore in
$S_{i_{0}}^{\prime}$, $(T_{C_{n}}^{k})^{-1}(R_{k}\cap C_{k})$ and that the
range of $T_{C^{n}}^{k+1}$ is the open set in $S_{i_{k+1}}^{\prime}$,
$T_{C_{k}}(R_{k}\cap C_{k})$. Note that $Z_{C^{n}}$ equals the domain of
$T_{C^{n}}^{n}$ which we have seen to be open. Using Proposition 3.7(3), we
have that $T_{C^{n}}^{k}(Z_{C^{n}})$ is open. By the above $T_{C^{n}}^{k}$ is
a homeomorphism on an open domain which contains $Z_{C^{n}}$ and therefore is
a homeomorphism on $Z_{C^{n}}$.

For Claim $(2)$, note $T_{C^{n}}^{-k}(V)=(T_{C^{n}}^{k})^{-1}(V\cap
T_{C^{n}}(Z_{C^{n}}))$ and $T_{C^{n}}(Z_{C^{n}})$ is open by (1). Thus $V\cap
T_{C^{n}}(Z_{C^{n}})$ is $F_{\sigma}$ and on this set $T_{C^{n}}^{-k}$ is a
homeomorphism. ∎

Using Proposition 3.8 let $\delta_{0}>0$ small enough so that $2\delta_{0}$ is
a Lebesgue number for each of the open covers $\mathcal{C}_{i,j}$ (of
$F_{i,j}$). Thus for all $1\leq i,j\leq N$,

(3)
$F_{i,j}\subset\bigcup_{V\in\mathcal{C}_{i,j}}\overline{\Theta}^{S_{i}^{\prime}}_{-\delta_{0}}(V).$

From now on in this section we abbreviate
$\overline{\Theta}_{\cdot}(\cdot)=\overline{\Theta}^{S_{1}^{\prime}}_{\cdot}(\cdot)$.

######  Definition 3.12.

We define

$Z_{C^{n}}^{\delta_{0}}:=\\{x\in\overline{\Theta}_{-\delta_{0}}(C_{0}):T_{C_{k-1}}\circ\cdots\circ
T_{C_{0}}(x)\in\overline{\Theta}_{-\delta_{0}}(C_{k})~{}\text{for}~{}1\leq
k\leq n\\}.$

This is clearly a closed set.

The following lemma will be important in the sequel:

######  Lemma 3.13.

Let $x\in S_{1}$ so that there are $0=t_{0}<t_{1}<t_{2}<\ldots<t_{d}$ with
$\Phi_{t_{k}}x\in S_{1}$ for $k=0,\ldots,d$. Then there exists
$C^{n}=(C_{0},C_{1},\dots,C_{n-1},C_{n})\in\mathcal{C}(n)$ for some
$n\in\mathbb{N}$, and $j_{0}=0<j_{1}<\ldots<j_{d}$ such that $x\in
Z_{C^{n}}^{\delta_{0}}$ and $T_{C^{n}}^{j_{i}}x=\Phi_{t_{i}}x$ for
$i=0,\ldots,d$.

###### Proof.

By Lemma 3.5, $x\in F_{1,j_{1}}$ for some $j_{1}$. By Equation (3), we may
find $C_{0}\in\mathcal{C}_{1,j_{1}}$ so that
$x\in\overline{\Theta}_{-\delta_{0}}(C_{0})$. A priori $T_{C_{0}}(x)\in
S_{j_{1}}^{\prime}$, however by Lemma 3.6(2), $T_{C_{0}}(x)\in S_{j_{1}}$.
Thus by Lemma 3.5, $T_{C_{0}}(x)\in F_{j_{1},j_{2}}$ for some $j_{2}$.
Repeating the argument so that we may find $C_{1}\in\mathcal{C}_{j_{1},j_{2}}$
so that $T_{C_{0}}(x)\in\overline{\Theta}_{-\delta_{0}}(C_{1})$. Continuing
inductively we construct a sequence $C^{n}:=(C_{0},C_{1},\ldots
C_{n})\in\mathcal{C}(n)$ with the properties:

  1. (1)

$x\in Z_{C^{n}}^{\delta_{0}}$.

  2. (2)

$T_{C^{n}}^{k}(x)=\Phi_{t_{\mathcal{G}}(T_{C^{n}}^{k-1}(x))}(T_{C^{n}}^{k-1}(x))$,
for $k=1,\ldots,n$.

  3. (3)

$n\gamma\geq t_{d}.$

Conditions (2) and (3) guarantee that there exist $j_{0}=0<j_{1}<\ldots<j_{d}$
such that $T_{C^{n}}^{j_{i}}x=\Phi_{t_{i}}x$ for $i=0,\ldots,d$.

∎

###  3.3. Statement of the Main Theorem

The main result is as follows.

######  Theorem 3.14 (=Theorem A).

Let $X$ be a finite-dimensional space. Let $\Phi$ be a topological flow on $X$
without fixed points, having a countable number of periodic orbits. Then
$(X,\Phi)$ has the small flow boundary property.

We follow the strategy of [Lin95]. Lindenstrauss proved that for any discrete
finite dimensional dynamical system $(X,T)$ with (arbitrary) periodic points
set $\operatorname{Per}(X)$, for every pair of open sets $U,V\subset X$, such
that $\partial U\setminus\operatorname{Per}(X)\subset V$, there is an open set
$U^{\prime}$ with $U\subset U^{\prime}\subset U\cup V$, such that $\partial
U^{\prime}$ is the union of a set of zero (discrete) orbit capacity and a
subset of $\operatorname{Per}(X)$. This statement should be compared with
Theorem 3.25. Our proof is not a routine generalization. Using the same
method, it is very probable one could prove an analog statement to the theorem
by Lindenstrauss, imposing no condition on periodic orbits.

Burguet [Bur19, Proposition 2.1] proved that a $C^{2}$-smooth flow without
fixed points on a compact smooth manifold (without boundary), satisfying that
for any $\tau>0$ the number of periodic orbits of period less than $\tau$ is
finite, has the small flow boundary property. A key tool in Burguet’s proof is
what he calls the _$n$ -transverse property_ ($\thickapprox$ the _$n$ -general
property_ defined later in this section). This property is established through
successive approximation. A crucial point for the approximation scheme to work
is the so-called _$C^{1}$ -stability of transversality_, that is if $M_{1}$
and $M_{2}$ are compact transverse smooth manifolds then any compact smooth
manifolds $\tilde{M}_{1}$ and $\tilde{M}_{2}$ which are sufficiently small
$C^{1}$-perturbations of $M_{1}$ and $M_{2}$ are transverse ([KH97, Corollary
A.3.17]. There are two difficulties in generalizing Burguet’s method to
topological flows. Firstly transversality is defined in the context of smooth
manifolds. Secondly, even if we consider a topological flow on a compact
manifold we are faced with the fact that transversality is not _$C^{0}$
-stable_.

Let us give a very rough overview of our proof. We are given $x\in\text{\rm
Int}^{\Phi}S$, where $S$ is a closed cross-section. We may find $U,V\subset S$
with $x\in\text{\rm Int}^{\Phi}U=U$, $\text{\rm Int}^{\Phi}V=V$ and
$\partial^{\Phi}U\subset V$. Our goal is to find a perturbation of $U$ in the
form of $U^{\prime}\subset S$ with $U\subset U^{\prime}\subset U\cup V$ and
$\partial^{\Phi}U^{\prime}\subset V$ such that $\operatorname{\text{\rm
ocap}_{\sharp}}(\partial^{\Phi}U^{\prime})=0$. Actually we establish the
stronger property that for every $y\in S$,

(4) $\sharp\\{t\geq 0:\Phi_{t}(y)\in\partial^{\Phi}U^{\prime}\\}\leq d$

Consider a small neighborhood of $x\in\partial^{\Phi}U^{\prime}$ and consider
the return profile of this neighborhood: i.e., how elements in this
neighborhood return to $\partial^{\Phi}U^{\prime}$. This can be represented by
certain intersections of certain images of this neighborhood. This enables one
do use the dimension of this intersection as a proxy to number of returns.
Indeed if the dimension drops below zero, the intersection is empty and no
further return is possible. The dimension drop mechanism is the key to the
proof and this is captured by the concept of general position explained in the
next subsection.

###  3.4. General position

By Lemma 2.10, a global closed cross-section in $X$ has dimension $d$. We will
therefore use the following definition:

######  Definition 3.15.

A collection $\mathcal{A}$ of subsets in a $d$-dimensional space is said to be
in general position if for every finite $\mathcal{B}\subset\mathcal{A}$, one
has that

$\dim(\cap_{C\in\mathcal{B}}C)\leq\max\\{-1,d-|\mathcal{B}|\\}$

The definition of general position is due to John Kulesza [Kul95]. To acquire
a better understanding, we quote the following sentences from Lindenstrauss
[Lin95]:

The motivation for this definition is that given a collection of
$(d-1)$-dimensional subsets of an $n$-dimensional space then generically any
two will have intersection with dimension less than $d-2$, etc.

######  Remark 3.16.

For the purposes of the proof the most important consequence of Definition
3.15 is that every finite sub-collection $\mathcal{B}\subset\mathcal{A}$ with
$d+1$ elements has empty intersection.

Recall the definition of $T_{C^{n}}^{-k}$ in Equation (2) and the definition
of $I_{C^{n}}$ in Definition 3.9 in Subsection 3.2.

######  Definition 3.17.

Let $n\geq 0$. A set $V\subset S_{1}$ is called $C^{n}$-general for
$C^{n}\in\mathcal{C}(n)$ if
$\left(T_{C^{n}}^{-k}(\partial^{\Phi}V)\right)_{k\in I_{C^{n}}}$ is in general
position. It is called n-general if it is $C^{n}$-general for every
$C^{n}\in\mathcal{C}(n)$. Moreover, we say that it is $C^{n}$-general (or
n-general) on a set $U$ if $\left(T_{C^{n}}^{-k}(\partial^{\Phi}W)\cap
U\right)_{k\in I_{C^{n}}}$ is in general position (or respectively for all
$C^{n}\in\mathcal{C}(n)$).

The following remark is easy to establish:

######  Remark 3.18.

If a set $V\subset S_{1}$ is $n$-general then it is $m$-general for all $0\leq
m\leq n$.

######  Lemma 3.19.

If $W\subset S_{1}$ with $\dim(\partial^{\Phi}W)\leq d-1$, then $W$ is
$0$-general.

###### Proof.

Note that for $C^{0}=(C_{0})\in\mathcal{C}(0)$, $I_{C^{0}}=\\{0\\}$. Therefore
$\dim(\partial^{\Phi}W)\leq d-1$ implies the result. ∎

###  3.5. Key Lemma

In this subsection, we prove the key technical lemma of the paper:

######  Lemma 3.20.

Let $U$ be a subset in $S_{1}$ with $\text{\rm Int}^{\Phi}U=U$ and $n\geq 0$.
Then for every subset $V$ in $S_{1}$ with $\text{\rm Int}^{\Phi}V=V$ and
$\partial^{\Phi}U\subset V$, there is a subset $U^{\prime}\subset S_{1}$ with
$\text{\rm Int}^{\Phi}U^{\prime}=U^{\prime}$ such that

  * (1)

$U\subset U^{\prime}\subset U\cup V$;

  * (2)

$\partial^{\Phi}U^{\prime}\subset V$;

  * (3)

$U^{\prime}$ is $n$-general;

  * (4)

$\partial^{\Phi}U^{\prime}\cap P=\emptyset$.

###### Proof of Lemma 3.20.

If $\partial^{\Phi}V=\emptyset$, then we set $U^{\prime}=U\cup V$. By Lemma
2.14 and the fact that $\partial^{\Phi}U\subset\text{\rm
Int}^{\Phi}(V)\subset\text{\rm Int}^{\Phi}(U\cup V)$, we see that
$\partial^{\Phi}U^{\prime}=\emptyset$. Thus $U^{\prime}$ is $n$-general for
all $n\geq 0$.

Let us assume $\partial^{\Phi}V\not=\emptyset$. We will prove the lemma by
induction on $n$. Note that by assumption the number of periodic orbits is
countable. As a periodic orbit intersects a cross-section a finite number of
times, we conclude $P\cap S_{1}$ is countable. By Theorem 2.25, the
$F_{\sigma}$ set $P\cap S_{1}$ is at most zero-dimensional. By Lemma 2.10,
$\dim(S_{1})\leq d$. By Theorem 2.29, one may find a $F_{\sigma}$ subset $E$
of $S_{1}$ with $\dim(E)\leq 0$ such that $\dim(S_{1}\setminus E)\leq d-1$. By
Corollary 2.26 $\dim(E\cup P\cap S_{1})\leq 0$. We now treat the base case
$n=0$. By Theorem 2.24 we may pick for every $x\in\partial^{\Phi}U$ an open
neighborhood $U_{x}\subset V$ such that $\partial^{\Phi}U_{x}\cap(P\cup
E)=\emptyset$. Let $U_{x_{1}},U_{x_{2}},\ldots,U_{x_{m}}$ be a cover of
$\partial^{\Phi}U$. Define $U^{\prime}=U\cup\bigcup_{i=1}^{m}U_{x_{i}}$.
Clearly
$\partial^{\Phi}U^{\prime}\subset\bigcup_{i=1}^{m}\partial^{\Phi}U_{x_{i}}$
and therefore Condition (4) holds. In addition it is easy to see that
Conditions (1) and (2) hold. As $\partial^{\Phi}U^{\prime}\subset
S_{1}\setminus E$, $\dim(\partial^{\Phi}U^{\prime})\leq d-1$. By Lemma 3.19,
Condition (3) holds.

By induction we suppose that we have constructed $A_{0}\subset S_{1}$ with
$\text{\rm Int}^{\Phi}A_{0}=A_{0}$ which satisfies the following properties:

  * (A1)

$U\subset A_{0}\subset U\cup V$.

  * (A2)

$\partial^{\Phi}A_{0}\subset V$.

  * (A3)

$A_{0}$ is $(n-1)$-general777Recall Remark 3.18..

  * (A4)

$\mathcal{C}(n)=\mathcal{F}_{0}\sqcup\mathcal{F}_{1}$ such that
$\mathcal{F}_{1}\not=\emptyset$ and $A_{0}$ is $C^{n}$-general for all
$C^{n}\in\mathcal{F}_{0}$.

  * (A5)

$\partial^{\Phi}A_{0}\cap P=\emptyset$.

Now fix $\mathcal{D}\in\mathcal{F}_{1}$. We will construct $U^{\prime}$ such
that

  * (B1)

$\text{\rm Int}^{\Phi}U^{\prime}=U^{\prime}$.

  * (B2)

$U\subset U^{\prime}\subset U\cup V$ and $\partial^{\Phi}U^{\prime}\subset V$.

  * (B3)

$U^{\prime}$ is $(n-1)$-general.

  * (B4)

$U^{\prime}$ is $C^{n}$-general for all
$C^{n}\in\mathcal{F}_{0}\sqcup\\{\mathcal{D}\\}$.

  * (B5)

$\partial^{\Phi}U^{\prime}\cap P=\emptyset$.

Assuming that such $U^{\prime}$ has been constructed, we may complete the
proof by inducting on $\mathcal{F}_{1}$, finally obtaining
$\mathcal{F}_{0}=\mathcal{C}(n)$. Thus it remains to construct $U^{\prime}$
that satisfies the conditions $(B1)-(B5)$ which will be achieved in the
sequel. Indeed set $U^{\prime}=A_{m}$ of Lemma 3.23. ∎

######  Definition 3.21.

Denote by $B(x,r)$ with $x\in S_{1}^{\prime}$ and $r>0$ the intersection of
$S_{1}^{\prime}$ and the open ball centered at $x$ with radius $r$. Also,
denote by $B^{\prime}$ the doubling ball of $B$, i.e. if $B=B(x,r)$ then
$B^{\prime}=B(x,2r)$.

######  Lemma 3.22.

There exists open sets $\\{B_{i}\\}_{i=1}^{m}$ satisfying the following
conditions:

  * (D1)

for any $C\in\mathcal{C}(n)$ and for any distinct $j,\ell\in I_{C}$, one has
that $T_{C^{n}}^{-j}(B_{i}^{\prime})\cap
T_{C^{n}}^{-\ell}(B_{i}^{\prime})=\emptyset$;

  * (D2)

$\\{B_{i}\\}_{i=1}^{m}$ is an open cover of $\partial^{\Phi}A_{0}$;

  * (D3)

$\bigcup_{i=1}^{m}B_{i}\subset V$.

###### Proof.

Let $x\in\partial^{\Phi}A_{0}$. We will now construct a ball around $x$,
$B_{x}=B(x,r)$ for some $r>0$, such that $B_{x}\subset V$ and for any
$C\in\mathcal{C}(n)$ and for any distinct $j,\ell\in I_{C}$, one has that
$T_{C^{n}}^{-j}(B_{x}^{\prime})\cap
T_{C^{n}}^{-\ell}(B_{x}^{\prime})=\emptyset$. Since $\mathcal{C}(n)$ is a
finite collection, it is sufficient to consider a fixed
$C^{n}=(C_{0},C_{1},\dots,C_{n-1},C_{n})\in\mathcal{C}(n)$ and fixed
$(j,\ell)\in I_{C^{n}}\times I_{C^{n}}$ with $j>\ell$ (as one may take the
intersection of the open balls associated to each $C^{n}$ and each $j>\ell$).
Assume for a contradiction that for every $p\in\mathbb{N}$ we may find an open
neighborhood $W_{p}$ in $S_{1}^{\prime}$ of $x$ with
$\operatorname{diam}(W_{p})<\frac{1}{p}$ and $w_{p}\in X$ such that $w_{p}\in
T_{C^{n}}^{-j}W_{p}\cap T_{C^{n}}^{-\ell}W_{p}$. Let
$z_{p}:=T_{C^{n}}^{j}w_{p},\,y_{p}:=T_{C^{n}}^{\ell}w_{p}\in W_{p}$. This
implies we have found sequences $(y_{p})_{p\in\mathbb{N}}$ and
$(z_{p})_{p\in\mathbb{N}}$ in $X$ converging to $x$ such that
$T_{D^{j-\ell}}^{j-\ell}y_{p}=z_{p}$, where
$D^{j-\ell}=(C_{\ell},\dots,C_{j})\in\mathcal{C}(j-\ell)$. Thus by Proposition
3.8 there exist $t_{p}\leq 2\eta(j-\ell)$ such that $\Phi_{t_{p}}y_{p}=z_{p}$.
W.l.o.g. $t_{p}\rightarrow t$, for $k\to\infty$ for some $t\leq
2\eta(j-\ell)$. We conclude that $\Phi_{t}x=x$ which is a contradiction to the
fact that $x\notin P$ as $\partial^{\Phi}A_{0}\cap P=\emptyset$. Let now
$\\{B_{i}\\}_{i=1}^{m}$ be a finite subcover of $\partial^{\Phi}A_{0}$
extracted from $\\{B_{x}\\}$. Clearly properties (D1)-(D3) hold. ∎

######  Lemma 3.23.

There exists sets $\\{A_{k}\\}_{k=1}^{m}$ such that for $k\geq 1$:

  * (E1)

$A_{k-1}\subset A_{k}$ and $A_{k}\setminus A_{k-1}\subset B_{k}^{\prime}$;

  * (E2)

$\text{\rm Int}^{\Phi}A_{k}=A_{k}$;

  * (E3)

$\partial^{\Phi}A_{k}\subset V$;

  * (E4)

$\partial^{\Phi}A_{k}\cap P=\emptyset$;

  * (E5)

$\partial^{\Phi}A_{k}\subset\bigcup_{i=1}^{m}B_{i}$;

  * (E6)

$A_{k}$ is $(n-1)$-general and $C^{n}$-general for all
$C^{n}\in\mathcal{F}_{0}$;

  * (E7)

$A_{k}$ is $\mathcal{D}$-general on $\cup_{1\leq j\leq k}B_{j}$.

###### Proof.

We will construct sets $\\{A_{k}\\}_{k=1}^{m}$ by induction on $k$. As $A_{0}$
is used as the base case we notice from properties (A1)-(A5) that $A_{0}$
satisfies (E2)-(E6) and (E7) (in an empty fashion).

Suppose that $(A_{i})_{i=0}^{k-1}$ have been already defined, satisfying the
listed proprieties. Define:

$\mathcal{A}_{C}=\\{T_{C}^{-j}(\partial^{\Phi}A_{k-1}):j\in I_{C}\\},$

for every $C\in\mathcal{C}(n-1)\cup\mathcal{F}_{0}\cup\\{\mathcal{D}\\}$. By
Theorem 2.29, for every $\mathcal{A}\subset\mathcal{A}_{C}$ with
$\bigcap\mathcal{A}\neq\emptyset$, there exists an $F_{\sigma}$ subset
$E_{\mathcal{A}}^{C}\subset Z_{C}$ of dimension zero such that

(5) $\dim(\bigcap\mathcal{A}\setminus
E_{\mathcal{A}}^{C})=\dim(\bigcap\mathcal{A})-1.$

Define

$E=\bigcup_{C\in\mathcal{C}(n-1)\cup\mathcal{F}_{0}\cup\\{\mathcal{D}\\},\mathcal{A}\subset\mathcal{A}_{C},j\in
I_{C}}T_{C}^{j}E_{\mathcal{A}}^{C},$

Clearly, $E$ is a zero-dimensional set in $S_{1}^{\prime}$ by Corollary 2.26
and Lemma 3.11(2). Note that by the inductive assumption
$\partial^{\Phi}A_{k-1}\subset\bigcup_{i=1}^{m}B_{i}$. Thus one may find
$\epsilon>0$ so that:

$\overline{\Theta}_{\epsilon}(\overline{B_{k}}\cap\partial^{\Phi}A_{k-1})\subset\
\bigcup_{i=1}^{m}B_{i},$

where recall from Subsection 2.1 that the closed $\epsilon$-tube around a
closed set $Q\subset S_{1}^{\prime}$ is the set
$\overline{\Theta}_{\epsilon}(Q):=\\{y\in
S_{1}^{\prime}|\,d(y,Q)\leq\epsilon\\}$. Clearly
$\overline{\Theta}_{\epsilon}(\overline{B_{k}})\subset B_{k}^{\prime}$ for
$\epsilon>0$ small enough. Thus
$\overline{\Theta}_{\epsilon}(\overline{B_{k}}\cap\partial^{\Phi}A_{k-1})\subset
B_{k}^{\prime}\cap\bigcup_{i=1}^{m}B_{i}$. Note $P\cap S_{1}$ is a
$F_{\sigma}$ set with $\dim(P\cap S_{1})\leq 0$. By Corollary 2.26,
$\dim(E\cup P\cap S_{1})\leq 0$. By Lemma 2.30, there exists a set $W$ with
$\text{\rm Int}^{\Phi}W=W$ such that

  * (F1)

$\overline{W}\subset B_{k}^{\prime}\cap\bigcup_{i=1}^{m}B_{i}$;

  * (F2)

$\overline{\Theta}_{\epsilon}(\overline{B_{k}}\cap\partial^{\Phi}A_{k-1})\subset
W$;

  * (F3)

$\partial^{\Phi}W\cap(E\cup P)=\emptyset$.

Let $A_{k}=A_{k-1}\cup W$. We will show that $A_{k}$ is the set we look for.
By (F1), Conditions (E1) and (E5) hold. Since $\text{\rm Int}^{\Phi}W=W$, (E2)
follows by Lemma 2.14. Note:

(6)
$\partial^{\Phi}A_{k}\subset\partial^{\Phi}A_{k-1}\cup\partial^{\Phi}W\subset\partial^{\Phi}A_{k-1}\cup\overline{W}\subset
V.$

This gives (E3). By (F3) and (E4) for $k-1$ (E4) holds for $k$. Using (6) and
(F3), we have (E7). It remains to check $(E6)$ and $(E7)$. We start by proving
$(E6)$. Fix $C\in\mathcal{C}(n-1)\cup\mathcal{F}_{0}$. Assume for a
contradiction that $A_{k}$ is not $C$-general. We can thus find $I\subset
I_{C}$ with

$\dim(\cap_{j\in I}T_{C}^{-j}\partial^{\Phi}A_{k})>\max\\{-1,d-\sharp I\\}.$

By Equation (6) it follows that

$\begin{split}\cap_{j\in I}T_{C}^{-j}\partial^{\Phi}A_{k}&\subset\cap_{j\in
I}\left(T_{C}^{-j}\partial^{\Phi}A_{k-1}\cup
T_{C}^{-j}\partial^{\Phi}W\right)\\\
&\subset\cup_{\alpha\in\\{0,1\\}^{I}}\left(\cap_{j\in
I}K_{j}^{\alpha_{j}}\right),\end{split}$

where $K_{j}^{0}=T_{C}^{-j}\partial^{\Phi}A_{k-1}$ and
$K_{j}^{1}=T_{C}^{-j}\partial^{\Phi}W$ for $j\in I$. By $(D1)$ and the fact
that $W\subset B_{k}^{\prime}$, we see that $\cap_{j\in
I}K_{j}^{\alpha_{j}}=\emptyset$ if $\alpha_{j}=1$ for at least two $j\in I$.
By the induction hypothesis, we have that $\dim(\cap_{j\in
I}K_{j}^{0})\leq\max\\{-1,d-\sharp I\\}$. By Lemmas 2.27 and 3.11 the sets
$K_{j}^{\alpha_{j}}$ are $F_{\sigma}$. As the intersection of finitely many
$F_{\sigma}$ sets is an $F_{\sigma}$ set, we may apply the sum theorem for
$F_{\sigma}$ sets (Corollary 2.26). It follows that there is $\ell\in I$ such
that

(7) $\dim(\big{(}\cap_{j\in I\setminus\\{\ell\\}}K_{j}^{0}\big{)}\cap
K_{\ell}^{1})>\max\\{-1,d-\sharp I\\}.$

Let $\mathcal{A}=\\{K_{j}^{0}\\}_{j\in I\setminus\\{\ell\\}}$. Assume there is
$x\in K_{\ell}^{1}\cap E_{\mathcal{A}}^{C}$, then
$T_{C}^{\ell}x\in\partial^{\Phi}W\cap T_{C}^{\ell}E_{\mathcal{A}}^{C}$ which
is a contradiction to $(F3)$. We conclude

$\big{(}\cap_{j\in I\setminus\\{\ell\\}}K_{j}^{0}\big{)}\cap
K_{\ell}^{1}\subset\big{(}\cap_{j\in
I\setminus\\{\ell\\}}K_{j}^{0}\big{)}\setminus E_{\mathcal{A}}^{C}.$

By the inductive hypothesis (E6) holds for $k-1$ and may be applied on
$\cap_{j\in I\setminus\\{\ell\\}}K_{j}^{0}$. Thus, using Equation (5):

$\begin{split}\dim(\big{(}\cap_{j\in I\setminus\\{\ell\\}}K_{j}^{0}\big{)}\cap
K_{\ell}^{1})&\leq\dim(\big{(}\cap_{j\in
I\setminus\\{\ell\\}}K_{j}^{0}\big{)}\setminus E_{\mathcal{A}}^{C})\\\
&=\dim(\cap_{j\in I\setminus\\{\ell\\}}K_{j}^{0})-1\\\ &\leq\max\\{-1,d-\sharp
I\\}.\end{split}$

This is a contradiction to Equation (7). We now prove (E7). We notice from
(F2) and the fact that $A_{k}=A_{k-1}\cup W$ that
$\partial^{\Phi}A_{k}\cap(B_{k}\cap\partial^{\Phi}A_{k-1})=\emptyset$,
equivalently
$\partial^{\Phi}A_{k}\setminus(B_{k}\cap\partial^{\Phi}A_{k-1})=\partial^{\Phi}A_{k}$.
In addition we conclude from (F1) that $\partial^{\Phi}W\setminus
B_{k}^{\prime}=\emptyset$. Thus
$\partial^{\Phi}A_{k}=\partial^{\Phi}A_{k}\setminus(B_{k}\cap\partial^{\Phi}A_{k-1})\subset(\partial^{\Phi}A_{k-1}\cup\partial^{\Phi}W)\setminus(B_{k}\cap\partial^{\Phi}A_{k-1})\subset(\partial^{\Phi}A_{k-1}\setminus
B_{k})\cup(\partial^{\Phi}W\cap B_{k}^{\prime})$. Finally:

$\bigcup_{i=1}^{k}B_{i}\cap\partial^{\Phi}A_{k}\subset(B_{k}\cup\bigcup_{i=1}^{k-1}B_{i})\cap\big{(}(\partial^{\Phi}A_{k-1}\setminus
B_{k})\cup(\partial^{\Phi}W\cap B_{k}^{\prime})\big{)}$

The (RHS) equals the union of the four following expressions, one of which is
empty:

  * •

$B_{k}\cap(\partial^{\Phi}A_{k-1}\setminus B_{k})=\emptyset$,

  * •

$\bigcup_{i=1}^{k-1}B_{i}\cap(\partial^{\Phi}A_{k-1}\setminus
B_{k})\subset\bigcup_{i=1}^{k-1}B_{i}\cap\partial^{\Phi}A_{k-1}$,

  * •

$B_{k}\cap\partial^{\Phi}W$

  * •

$(\bigcup_{i=1}^{k-1}B_{i})\cap(\partial^{\Phi}W\cap B_{k}^{\prime})$

Fix $I\subset I_{\mathcal{D}}$. One can write $\bigcap_{j\in
I}T_{\mathcal{D}}^{-j}(\bigcup_{i=1}^{k}B_{i}\cap\partial^{\Phi}A_{k})$ as the
union of $3^{|I|}$ expressions. Notice that as above each one of these
expressions is a finite intersection of $F_{\sigma}$ sets, thus $F_{\sigma}$
by itself. Therefore by Corollary 2.26, it is enough to show that each of
these expressions has dimension less or equal than $d-|I|$. Let us analyze
each of these expressions. If the expression $B_{k}\cap\partial^{\Phi}W$
appears twice, then the expression is empty as
$T_{\mathcal{D}}^{-j_{1}}(B_{k})\cap
T_{\mathcal{D}}^{-j_{2}}(B_{k})=\emptyset$ for $j_{1}\neq j_{2}$. The same is
true for the expression $(\bigcup_{i=1}^{k-1}B_{i})\cap(\partial^{\Phi}W\cap
B_{k}^{\prime})$ as $T_{\mathcal{D}}^{-j_{1}}(B_{k}^{\prime})\cap
T_{\mathcal{D}}^{-j_{2}}(B_{k}^{\prime})=\emptyset$ for $j_{1}\neq j_{2}$.
Thus the expressions involving $\partial^{\Phi}W$ appear at most once. If such
expression do not appear at all, then we are only left with the expression
$\bigcup_{i=1}^{k-1}B_{i}\cap\partial^{\Phi}A_{k-1}$ and the dimension
estimate is handled by the inductive assumption that condition (E7) holds for
$k-1$. Finally we treat the case where an expression involving
$\partial^{\Phi}W$ appears exactly once. We thus analyze an expression
contained in an expression of the form

$T_{\mathcal{D}}^{-\ell}\partial^{\Phi}W\cap\bigcap_{j\in{I\setminus\\{\ell\\}}}T_{\mathcal{D}}^{-j}(\bigcup_{i=1}^{k-1}B_{i}\cap\partial^{\Phi}A_{k-1})$

Let
$\mathcal{A}=\\{T_{\mathcal{D}}^{-j}(\partial^{\Phi}A_{k-1}):j\in{I\setminus\\{\ell\\}}\\}$,
Assume there is $x\in T_{\mathcal{D}}^{-\ell}\partial^{\Phi}W\cap
E_{\mathcal{A}}^{\mathcal{D}}$, then
$T_{\mathcal{D}}^{\ell}x\in\partial^{\Phi}W\cap
T_{\mathcal{D}}^{\ell}E_{\mathcal{A}}^{\mathcal{D}}\subset\partial^{\Phi}W\cap
E$ which is a contradiction to $(F3)$. Thus, using condition (E7) for $k-1$
and Equation (5),

$\begin{split}&\dim(T_{\mathcal{D}}^{-\ell}\partial^{\Phi}W\cap\bigcap_{j\in{I\setminus\\{\ell\\}}}T_{\mathcal{D}}^{-j}(\bigcup_{i=1}^{k-1}B_{i}\cap\partial^{\Phi}A_{k-1}))\\\
&\leq\dim(\big{(}\bigcap_{j\in{I\setminus\\{\ell\\}}}T_{\mathcal{D}}^{-j}(\bigcup_{i=1}^{k-1}B_{i}\cap\partial^{\Phi}A_{k-1})\big{)}\setminus
E_{\mathcal{A}}^{\mathcal{D}})\\\
&=\dim(\bigcap_{j\in{I\setminus\\{\ell\\}}}T_{\mathcal{D}}^{-j}(\bigcup_{i=1}^{k-1}B_{i}\cap\partial^{\Phi}A_{k-1}))-1\\\
&\leq\max\\{-1,d-|I|\\},\end{split}$

as desired.

∎

###  3.6. Proof of the Main Theorem

The proof of the main theorem necessitates an approximation lemma. Recall the
definition of $\delta_{0}$ in Definition 3.12.

######  Lemma 3.24.

Let $A$ and $V$ be subsets such that $A=\overline{A}\subset V=\text{\rm
Int}^{\Phi}V\subset\text{\rm Int}^{\Phi}S_{1}$. Let $n,d\in\mathbb{N}$.
Suppose that for all $C^{n}=(C_{0},C_{1},\ldots,C_{n})\in\mathcal{C}(n)$ and
$I\subset I_{C}$ with $0\in I$ and $\sharp I=d+1$, it holds,

(8) $\bigcap_{k\in I}T_{C^{n}}^{-k}A=\emptyset,$

Then there exists $U\subset\overline{U}\subset V\subset S_{i}$ with $\text{\rm
Int}^{\Phi}U=U$ and $A\subset U$ such that

(9) $Z_{C^{n}}^{\delta_{0}}\cap\bigcap_{k\in I}T_{C^{n}}^{-k}U=\emptyset,$

for all $C^{n}=(C_{0},C_{1},\ldots,C_{n})\in\mathcal{C}(n)$ and $I\subset
I_{C}$ with $0\in I$ and $\sharp I=d+1$.

###### Proof.

We claim there is $m_{0}\in\mathbb{N}$, so that for all $m\geq m_{0}$,
$U=U_{m}:=\Theta_{1/m}(A)$ fulfills the sought after requirements. Assume for
a contradiction that this is not the case. We may thus find $x_{m}\in
Z_{C_{m}^{n}}^{\delta_{0}}$,
$C_{m}^{n}=(C_{0,m},C_{1,m},\ldots,C_{n,m})\in\mathcal{C}(n)$,
$0=j_{0}^{m}<j_{1}^{m}<\cdots<j_{d}^{m}$, $\\{j_{i}^{m}\\}_{i=0}^{d}\subset
I_{C_{m}^{n}}$ so that $T_{C_{m}^{n}}^{j_{i}^{m}}x_{m}\in\Theta_{1/m}(A)$ for
$i=0,\ldots,d$. As $\mathcal{C}(n)$ is finite, by passing to a subsequence we
may assume w.l.o.g. $C_{m}^{n}=C^{n}$ for some fixed
$C^{n}=(C_{0},C_{1},\ldots,C_{n})\in\mathcal{C}(n)$, and $j_{i}^{m}=j_{i}$ for
$i=0,\ldots,d$, for some fixed $I:=\\{j_{i}\\}_{i=0}^{d}\subset I_{C^{n}}$. By
passing to a subsequence we may assume w.l.o.g. that $x_{m}\rightarrow x\in
Z_{C^{n}}^{\delta_{0}}$. In particular $T_{C^{n}}^{j_{i}}x$ is well defined
for $i=0,\ldots,d$. Conclude $T_{C^{n}}^{j_{i}}x\in A$ for $i=0,\ldots,d$.
Thus $x\in\bigcap_{k\in I}T_{C^{n}}^{-k}A$ which is a contradiction. Q.E.D. ∎

######  Theorem 3.25.

Let $X$ be a compact finite-dimensional space. Let $\Phi$ be a topological
flow on $X$ without fixed points, having a countable number of periodic
orbits. Then for any $U,V\subset S_{1}$ with $\text{\rm Int}^{\Phi}U=U$,
$\text{\rm Int}^{\Phi}V=V$ and $\partial^{\Phi}U\subset V$, there exists
$U^{\prime}\subset S_{1}$ with $\text{\rm Int}^{\Phi}U^{\prime}=U^{\prime}$
and $U\subset U^{\prime}\subset U\cup V$ such that $U^{\prime}$ has a small
flow boundary.

###### Proof.

By induction, we will construct $\\{U_{k}\\}_{k=0}^{\infty}$ and
$\\{V_{k}\\}_{k=0}^{\infty}$ satisfying for all $k\geq 0$,

  * (1)

$U_{0}=U$ and $V_{0}=V$;

  * (2)

$\text{\rm Int}^{\Phi}U_{k}=U_{k}\subset S_{1}$ and $\text{\rm
Int}^{\Phi}V_{k}=V_{k}\subset S_{1}$;

  * (3)

$\partial^{\Phi}U_{k}\subset V_{k}$;

  * (4)

$\overline{V_{k+1}}\subset V_{k}$ and $U_{k}\subset U_{k+1}$;

  * (5)

$U_{k+1}\subset U_{k}\cup V_{k}$;

  * (6)

for every $C^{k}=(C_{0},C_{1},\ldots,C_{k})\in\mathcal{C}(k)$, one has that
$Z_{C^{n}}^{\delta_{0}}\cap\bigcap_{j\in I}T_{C^{k}}^{-j}V_{k}=\emptyset$ for
all $I\subset I_{C^{k}}$ with $0\in I$ and $\sharp I=d+1$.

Assuming that such $\\{U_{k}\\}_{k=0}^{\infty}$ and
$\\{V_{k}\\}_{k=0}^{\infty}$ have been constructed, we set
$U^{\prime}=\cup_{i=0}^{\infty}U_{i}$. It is clear that $U^{\prime}$ is
relatively open in $S_{1}^{\prime}$. By $(4)$ and $(5)$, we have that

$U_{k+\ell}\cup V_{k}\subset U_{k+\ell-1}\cup V_{k+\ell-1}\cup V_{k}\subset
U_{k+\ell-1}\cup V_{k},$

for all $k,\ell>0$. It follows that

$U_{k+\ell}\cup V_{k}\subset U_{k}\cup V_{k},$

and thus as by (4) $U^{\prime}=\bigcup_{\ell\geq k}U_{\ell}$ for all $k$,

$U_{k+1}\subset U^{\prime}\subset U_{k+1}\cup V_{k+1},$

for all $k>0$. Then by $(2),(3),(4)$ we obtain that

(10) $\partial^{\Phi}U^{\prime}\subset\overline{U_{k+1}\cup V_{k+1}}\setminus
U_{k+1}\subset\partial^{\Phi}U_{k+1}\cup\overline{V_{k+1}}\subset V_{k},$

for all $k>0$. In order to establish $\operatorname{\text{\rm
ocap}_{\sharp}}(\partial^{\Phi}U^{\prime})=0$, it is sufficient to show that
any orbit of the flow hits at most $d$ times the set
$\partial^{\Phi}U^{\prime}$. Let $x\in\partial^{\Phi}U^{\prime}\subset S_{1}$
and assume for a contradiction that there are $0<t_{1}<t_{2}<\ldots<t_{d}$
such that $\Phi_{t_{i}}x\in\partial^{\Phi}U^{\prime}\subset S_{1}$,
$i=1,\ldots,d$.

By Lemma 3.13, for some $n$, there is
$C^{n}=(C_{0},C_{1},\dots,C_{n})\in\mathcal{C}(n)$ with $x\in
Z_{C^{n}}^{\delta_{0}}$ and $I=\\{j_{i}\\}_{i=0}^{d}\subset I_{C^{n}}$ with
$j_{0}=0<j_{1},\ldots<j_{d}$ such that $T_{C^{n}}^{j_{i}}x=\Phi_{t_{i}}x$ for
$i=0,\ldots,d$. Thus

$x\in Z_{C^{n}}^{\delta_{0}}\cap\bigcap_{j\in
I}T_{C^{n}}^{-j}\partial^{\Phi}U^{\prime}\subset
Z_{C^{n}}^{\delta_{0}}\cap\bigcap_{j\in I}T_{C^{n}}^{-j}V_{n}\neq\emptyset,$

which is a contradiction to $(6)$.

It remains to construct $\\{U_{k}\\}_{k=0}^{\infty}$ and
$\\{V_{k}\\}_{k=0}^{\infty}$. By (1), the sets $U_{0}$ and $V_{0}$ are well
defined for $k=0$. Suppose that $(U_{i})_{i=0}^{k}$ and $(V_{i})_{i=0}^{k}$
have been constructed. Applying Lemma 3.20 to $U_{k},V_{k}$, we have a subset
$U_{k+1}\subset S_{1}$ with $\text{\rm Int}^{\Phi}U_{k+1}=U_{k+1}$ such that

  * (A1)

$U_{k}\subset U_{k+1}\subset U_{k}\cup V_{k}$;

  * (A2)

$\partial^{\Phi}U_{k+1}\subset V_{k}$;

  * (A3)

$U_{k+1}$ is $(k+1)$-general.

Applying Lemma 3.24 and Remark 3.16 to $A=\partial^{\Phi}U_{k+1}$, we have a
set $V_{k+1}\subset S_{1}$ with $\text{\rm Int}^{\Phi}V_{k+1}=V_{k+1}$ such
that

  * (B1)

$\partial^{\Phi}U_{k+1}\subset V_{k+1}\subset\overline{V_{k+1}}\subset V_{k}$;

  * (B2)

for all $C^{k+1}=(C_{0},\ldots,C_{k+1})\in\mathcal{C}(k+1)$, it holds that
$Z_{C^{n}}^{\delta_{0}}\cap\bigcap_{j\in I}T_{C^{k+1}}^{-j}V_{k+1}=\emptyset$
for all $I\subset I_{C^{k+1}}$ with $0\in I$ and $\sharp I=d+1$.

Note that the set $U_{k+1}$ and $V_{k+1}$ satisfy the conditions $(2)-(6)$. ∎

We can now prove the main theorem:

###### Proof of Theorem 3.14.

Fix a point $x\in X$ and a closed cross-section $S$ with $x\in\text{\rm
Int}^{\Phi}S$. Using Lemma 5.1, one may find two complete families of cross-
sections $\SS=\\{S_{i}\\}_{i=1}^{N}$ and
$\SS^{\prime}=\\{S_{i}^{\prime}\\}_{i=1}^{N}$ obeying the conditions in the
beginning of Section 3.1, with $x\in\text{\rm Int}^{\Phi}(S_{1})\subset
S_{1}\subset S$. We now choose $U$ and $V$ in $S_{1}$ with $\text{\rm
Int}^{\Phi}U=U$ and $\text{\rm Int}^{\Phi}V=V$ such that $x\in U\subset S_{1}$
and $\partial^{\Phi}U\subset V\subset S_{1}$. Applying Theorem 3.25 to $U$ and
$V$, we have an open set $U^{\prime}\subset S_{1}$ with $\text{\rm
Int}^{\Phi}U^{\prime}=U^{\prime}$ and $U\subset U^{\prime}\subset U\cup V$
such that $U^{\prime}$ has a small flow boundary and $x\in U^{\prime}$. We
thus conclude that $(X,\Phi)$ has the small flow boundary property.

∎

##  4\. Expansive flows have strongly isomorphic symbolic extensions

###  4.1. Background

In this section, we are interested in expansive flows. We first recall the
definition of expansive flows and several properties of such flows. Then we
build the symbolic extension of an expansive flow following Bowen and Walters.
Finally we give a positive answer to an open question by Bowen and Walters,
showing that any expansive topological flow has a strongly isomorphic symbolic
extension.

The system $(X,T)$ is said to be expansive if there exists an $\delta>0$ such
that $d(T^{n}x,T^{n}y)<\delta$ for all $n\in\mathbb{Z}$ implies $x=y$ where
$d$ is a metric on $X$. For example, Anosov diffeomorphisms are expansive (see
[Ano67]). Keynes and Robertson [KR69] proved that every expansive
homeomorphism has a symbolic extension. Moreover, when $X$ is $0$-dimensional,
it is conjugate to a subshift. The expansive systems have been extensively
investigated and many interesting properties have been proven. For example,
expansiveness is invariant under topological conjugacy; the periodic
growth888The periodic growth of $(X,T)$ is defined as
$\limsup_{n\rightarrow\infty}(1/n)\log|P_{n}|$ where $P_{n}$ is the set of
$n$-periodic points. and the topological entropy of an expansive discrete
system is finite [Con68]. Mañé [Mañ79] showed that an expansive
$\mathbb{Z}$-system must be finite-dimensional. From this it follows that an
expansive $\mathbb{Z}$-system without fixed points has the small boundary
property ([Lin95]). By [Bur19, Proposition C.1] an expansive
$\mathbb{Z}$-system with the small boundary property admits an essential
uniform generator which induces a strongly isomorphic symbolic extension
([Bur19, Proposition 3.1]).

######  Definition 4.1.

The flow $(X,\Phi)$ is said to be expansive if for any $\epsilon>0$, there
exists $\delta>0$ such that if $d(\Phi_{t}(x),\Phi_{s(t)}y)<\delta$ for all
$t\in\mathbb{R}$, for a pair of points $x,y\in X$ and a continuous map
$s:\mathbb{R}\to\mathbb{R}$ with $s(0)=0$, then then there exists
$t\in\mathbb{R}$ with $|t|<\epsilon$ such that $y=\Phi_{t}(x)$.

The notion of expansiveness of topological flows was introduced by Bowen and
Walters [BW72]. We remark that the definition of expansiveness is clearly
independent of the metric. By [BW72, Theorem 5], the entropy of an expansive
flow is finite and for every $t\geq 0$ the number of periodic orbits of period
in $[0,t]$ is finite. By [BW72, Lemma 1], an expansive flow has only a finite
number of fixed points and each of them is an isolated point999Note that an
expansive $\mathbb{Z}$-system has also only a finite number of fixed points.
However, these points need not be isolated as witnessed by the (expansive)
full shift on two letters. . This reduces the study of expansive flows to
those without fixed points. Another important property of expansiveness which
Bowen and Walters proved is that expansiveness is a conjugacy invariant for
flows [BW72, Corollary 4]. One may therefore argue that the definition of
expansiveness by Bowen and Walters is the correct one (see also the discussion
of other attempts in [BW72, Page 180-181]).

Keynes and Sears [KS81] extended the above mentioned result of Mañé, showing
that expansive flows must be finite-dimensional. It can be shown that Anosov
flows are expansive ([Ano67]). The geodesic flow on a compact smooth manifold
of negative curvature is an Anosov flow and thus expansive; also, Axiom A
flows are expansive on their nonwandering sets [Bow72].

###  4.2. Bowen and Walters’ construction

In this section, we recall the construction of Bowen and Walters, i.e. the
symbolic extension of an expansive flow. Let $(X,\Phi)$ be an expansive flow
without fixed points. The following exposition is based on [BW72, Section 5].

######  Theorem 4.2 ([BW72], Theorem 3).

A topological flow without fixed points $(X,\Phi)$ is expansive if and only if
for any $\epsilon>0$ there exists $\alpha=\alpha(\epsilon)>0$ such that the
following holds: if $\mathbf{t}=(t_{i})_{i\in\mathbb{Z}}$ and
$\mathbf{u}=(u_{i})_{i\in\mathbb{Z}}$ with $t_{0}=u_{0}=0$,
$0<t_{i+1}-t_{i}\leq\alpha$, $|u_{i+1}-u_{i}|\leq\alpha$, $t_{i}\to\infty$ and
$t_{-i}\to-\infty$ as $i\to\infty$ and if $x,y\in X$ satisfy
$d(\Phi_{t_{i}}(x),\Phi_{u_{i}}(y))\leq\alpha$ for all $i\in\mathbb{Z}$, then
there exists $t\in\mathbb{R}$ with $|t|\leq\epsilon$ such that
$y=\Phi_{t}(x)$.

Let $\eta$ be as in Lemma 5.1. W.l.o.g. we assume $\alpha<\eta$. Choose
$\alpha=\alpha(\eta)>0$ as in Theorem 4.2. By Lemma 5.1, we can find a
complete family $\mathcal{S}=\\{S_{i}\\}_{i=1}^{N}$ of injectivity time
$\eta>0$ with $\Phi_{[0,\alpha]}\mathcal{G}=\Phi_{[-\alpha,0]}\mathcal{G}=X$,
where $\mathcal{G}=\cup_{1\leq i\leq N}S_{i}$. Since $S_{i}$ are pairwise
disjoint and compact, there is $\beta>0$ so that $\Phi_{[0,\beta]}(S_{i})$ are
pairwise disjoint for all $i$. For $x\in\mathcal{G}$, let $r_{i}(x)$ be the
$i$-th return time to $\mathcal{G}$ (see Definition 3.2). As
$\Phi_{[-\alpha,0]}\mathcal{G}=X$, every $y\in\mathcal{G}$, has some
$x\in\mathcal{G}$ and $0\leq s\leq\alpha$ so that $\Phi_{-s}x=y$, i.e.
$\Phi_{s}y=x\in\mathcal{G}$. It follows that $0<\beta\leq
r_{i+1}(x)-r_{i}(x)\leq\alpha$ for all $x\in\mathcal{G}$. Denote by
$\Sigma=\\{1,2,\dots,N\\}$ and $\sigma$ the (left) shift on
$\Sigma^{\mathbb{Z}}$. Define $\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}$ by

$\\{(x,\mathbf{q})\in\mathcal{G}\times\Sigma^{\mathbb{Z}}:\exists\mathbf{t}\in\mathbb{R}^{\mathbb{Z}}~{}\text{so
that}~{}t_{0}=0,t_{i+1}-t_{i}\in[\beta,\alpha],\Phi_{t_{i}}(x)\in
S_{q_{i}}\\}.$

In the definition $\mathbf{t}$ is called a return-time sequence for
$(x,\mathbf{q})$; $\mathbf{q}$ is called a return-name sequence for $x$ and
$x$ is called a realization of $\mathbf{q}$. A return-time sequence
$\mathbf{t}$ for $x$ is any return-time sequence of
$(x,\mathbf{q})\in\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}$ for some
$\mathbf{q}$.

Note that the sequence $\mathbf{r}=\\{r_{i}(x)\\}_{i=-\infty}^{\infty}$ is the
maximal return-time sequence for $x$ in the sense that if
$\mathbf{t}=\\{t_{i}\\}_{i=-\infty}^{\infty}$ is a return-time for $x$, then
$\\{t_{i}\\}_{i=-\infty}^{\infty}\subset\\{r_{i}(x)\\}_{i=-\infty}^{\infty}$.
Other return-time sequences than $\mathbf{r}$ for $x$ may exists. This is the
case if for example for some $i$, $r_{i+1}-r_{i-1}\leq\alpha$, as one can
remove $r_{i}$ from $\mathbf{r}$ and still have a return-time sequence for
$x$. In contrast Lemma 4.4 shows that $(x,\mathbf{q})$ has a unique return-
time sequence. In addition in Lemma 4.6 it is shown that the return-name
sequence determines a realization uniquely.

######  Lemma 4.3.

$\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}$ is closed in
$\mathcal{G}\times\Sigma^{\mathbb{Z}}$.

###### Proof.

Let $(x^{n},\mathbf{q}^{n})\in\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}$ which
converge to $(x,\mathbf{q})$ as $n\to\infty$. Let $\mathbf{t}^{n}$ be a
return-time sequence of $(x^{n},\mathbf{q}^{n})$. Since
$|t_{i}^{n}|\leq|i|\alpha$ for every $i\in\mathbb{Z}$, we may assume that the
limit $\lim_{n\to\infty}t_{i}^{n}$ exists, say $t_{i}$ for each
$i\in\mathbb{Z}$. Since $\Sigma$ is finite, we see that for each
$i\in\mathbb{Z}$, $q_{i}^{n}=q_{i}$ for large $n>n(i)$. Since $S_{q_{i}}$ is
closed for each $i\in\mathbb{Z}$, we obtain that
$\Phi_{t_{i}}(x)=\lim_{n\to\infty}\Phi_{t_{i}^{n}}(x^{n})\in S_{q_{i}}$ for
each $i\in\mathbb{Z}$. This implies that
$(x,\mathbf{q})\in\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}$ with return-time
sequence $\mathbf{t}=(t_{i})_{i\in\mathbb{Z}}$. ∎

######  Lemma 4.4.

For each $(x,\mathbf{q})\in\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}$, the return-
time sequence $\mathbf{t}$ is unique.

###### Proof.

Suppose that $\mathbf{t}$ and $\mathbf{t}^{\prime}$ are two distinct return-
time sequences of $(x,\mathbf{q})\in\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}$.
Without loss of generality, we assume that
$(t_{i})_{i>0}\not=(t_{i}^{\prime})_{i>0}$. Let
$k=\min\\{i>0:t_{i}\not=t_{i}^{\prime}\\}$. It follows that
$t_{k-1}=t_{k-1}^{\prime}$ and
$\Phi_{t_{k}}(x),\Phi_{t_{k}^{\prime}-t_{k}}(\Phi_{t_{k}}(x))\in S_{q_{k}}$.
Since $t_{k}-t_{k-1}\leq\alpha$ and
$t_{k}^{\prime}-t_{k-1}^{\prime}\leq\alpha$, we have that
$|t_{k}^{\prime}-t_{k}|\leq\alpha<\eta$ and consequently
$t_{k}=t_{k}^{\prime}$. This is a contradiction. ∎

Thus we may define the map:

$\mathbf{t}:\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}\to\mathbb{R}^{\mathbb{Z}}.$

######  Lemma 4.5.

The map $\mathbf{t}$ is continuous.

###### Proof.

It suffices to show that
$t_{i}:\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}\to[-|i|\alpha,|i|\alpha]\subset\mathbb{R}$
is continuous for each $i\in\mathbb{Z}$. Assume
$(x^{n},\mathbf{q}^{n})\in\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}$ converge to
$(x,\mathbf{q})$ as $n\to\infty$. Let
$\\{t_{i}(x^{n_{j}},\mathbf{q}^{n_{j}})\\}_{j}$ be an arbitrary converging
subsequence. By the argument of Lemma 4.3,
$\lim_{j\rightarrow\infty}t_{i}(x^{n_{j}},\mathbf{q}^{n_{j}})=t_{i}(x,\mathbf{q})$.
This establishes continuity. ∎

Let $P_{2}:\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}\to\Sigma^{\mathbb{Z}}$ be the
projection.

######  Lemma 4.6.

The map $P_{2}$ is injective.

###### Proof.

Suppose that $(x,{\bf q}),(y,{\bf
q})\in\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}$. Since $(X,\Phi)$ is expansive,
by Theorem 4.2, we see that $x=\Phi_{t}(y)$ for some $t$ with $|t|\leq\eta$.
Since $x,y\in S_{q_{0}}$ and $S_{q_{0}}$ is a cross-section of injectivity
time $\eta$, we have that $x=y$. ∎

Let $\mathcal{Z}:=\cup_{1\leq i\leq N}\partial^{\Phi}(S_{i})$ and
$\mathcal{W}$ be the set of points whose orbit does not intersect
$\mathcal{Z}$. In other words,

(11) $\mathcal{W}=X\setminus\bigcup_{r\in\mathbb{R}}\Phi_{r}(\mathcal{Z}).$

Clearly, the set $\mathcal{W}$ is $\Phi$-invariant, i.e.
$\Phi_{t}(\mathcal{W})=\mathcal{W}$ for all $t\in\mathbb{R}$. Define

$\mathcal{V}=\mathcal{W}\cap\mathcal{G}.$

Recall the definition of $r_{i}(x)$ above. Define
$Q:\mathcal{V}\to\Sigma^{\mathbb{Z}}$ by $Q(x)=(Q_{i}(x))_{i\in\mathbb{Z}}$
where $Q_{i}(x)=j$ if $\Phi_{r_{i}(x)}(x)\in S_{j}$. Note that the map $Q$ is
not necessarily continuous. If $x\in\mathcal{V}$, then also
$\Phi_{r_{1}(x)}(x)\in\mathcal{V}$, thus $Q(\mathcal{V})$ is
$\sigma$-invariant. Define

$\Lambda=\overline{Q(\mathcal{V})}.$

Clearly $(\Lambda,\sigma)$ is a subshift. Let
$P_{1}:\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}\to\mathcal{G}$ be the projection
on the first coordinate. Note that the map
$P_{2}^{-1}:P_{2}(\mathcal{G}\rtimes\Sigma^{\mathbb{Z}})\to\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}$
is continuous. Clearly, for $x\in\mathcal{V}$, the point
$(x,Q(x))\in\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}$ and consequently $Q(x)\in
P_{2}(\mathcal{G}\rtimes\Sigma^{\mathbb{Z}})$. Since
$P_{2}(\mathcal{G}\rtimes\Sigma^{\mathbb{Z}})$ is compact and contains
$Q(\mathcal{V})$, we see that $\Lambda$, which is the closure of
$Q(\mathcal{V})$, is contained in
$P_{2}(\mathcal{G}\rtimes\Sigma^{\mathbb{Z}})$. We define the continuous map
$P:\Lambda\to\mathcal{G}$ by

$P({\bf q})=P_{1}(P_{2}^{-1}({\bf q}))$

for ${\bf q}\in\Lambda$. Thus, $P$ associates with a return-name sequence in
$\Lambda$ its realization in $\mathcal{G}\subset X$. Clearly, $P\circ Q$ is
the identity on $\mathcal{V}$.

We define the continuous function $f:\Lambda\to[\beta,\alpha]$ by

$f({\bf q})=t_{1}(P_{2}^{-1}{\bf q}).$

######  Remark 4.7.

From the proof of Lemma 4.4 it is clear that $f({\bf q})$ is the first
positive time when the realization of $\bf q$ in $\mathcal{G}$ returns to
$S_{q_{1}}$ but it is possible that it returns to $\mathcal{G}$ earlier.
However, if $\bf q\in Q(\mathcal{V})$ then $f({\bf q})$ is the first positive
return time to $\mathcal{G}$ of the realization of $\bf q$ (due to the
definition of $Q$).

######  Definition 4.8.

Denote by $(\Lambda_{f},\Psi)$ the suspension flow over $(\Lambda,\sigma)$.
Define $\pi:\Lambda_{f}\to X$ by

$\pi(({\bf q},t))=\Phi_{t}(P({\bf q})).$

######  Theorem 4.9 (Theorem 10, [BW72]).

The flow $(X,\Phi)$ is a factor of $(\Lambda_{f},\Psi)$ via $\pi$.

###### Proof.

First, we show that $\pi$ is well defined, i.e. that it holds $\pi({\bf
q},f({\bf q}))=\pi(\sigma({\bf q}),0)$. This implies easily that $\pi$ is
$\mathbb{R}$-equivariant. By the definition of $\pi$, one has to show
$\pi(({\bf q},f({\bf q})))=\Phi_{f({\bf q})}(P({\bf q}))=P(\sigma({\bf
q}))=\pi((\sigma({\bf q}),0))$. It is enough to show $(\Phi_{f({\bf
q})}(P({\bf q})),\sigma({\bf q}))\in\mathcal{G}\rtimes\Sigma^{\mathbb{Z}}$ as
the return-name sequence determines the realization. Indeed, we claim that the
associated return-time sequence of $(\Phi_{f({\bf q})}(P({\bf q})),\sigma({\bf
q}))$ is $\\{{t}_{i+1}(P({\bf q}),{\bf q})-t_{1}(P({\bf q}),{\bf
q})\\}_{i\in\mathbb{Z}}$, which is confirmed by:

$\Phi_{{t}_{i+1}(P({\bf q}),{\bf q})-t_{1}(P({\bf q}),{\bf q})}(\Phi_{f({\bf
q})}(P({\bf q})))=\Phi_{{t}_{i+1}(P({\bf q}),{\bf q})}(P({\bf q}))\in
S_{q_{i+1}},$

for $i\in\mathbb{Z}$, where we used that $t_{1}(P({\bf q}),{\bf q})=f({\bf
q})$ (see Remark 4.7).

Finally, we note that the image $\pi(\Lambda_{f})$ is a $\Phi$-invariant
closed subset of $X$ which contains $\mathcal{V}$ as for $x\in\mathcal{V}$ it
holds $\pi((Q(x),0))=P\circ Q(x)=x$.

From Definition 2.7 it is easy to see that the closed set
$\Phi_{[-\eta,\eta]}\mathcal{Z}$ has empty interior (see also [Bur19, Lemma
2.4(1)]). By Baire category theorem this implies that
$\mathcal{W}=\bigcup_{r\in\mathbb{R}}\Phi_{r}(\mathcal{V})$ is dense in $X$.
We conclude that $\pi(\Lambda_{f})=X$. ∎

The following lemma is crucial for the proof of Theorem 4.11 in the sequel.

######  Lemma 4.10.

Let $({\bf q},t)\in\Lambda_{f}$ for some $0\leq t<f({\bf q})$. If $\pi(({\bf
q},t))=z\in\mathcal{V}$, then $t=0$ and ${\bf q}=Q(z)$.

###### Proof.

By definition $\pi(\Lambda_{f}\times\\{0\\})\subset\mathcal{G}$. Thus
$x:=\Phi_{-t}(z)\in\mathcal{G}\cap\mathcal{W}=\mathcal{V}$. Moreover $x=P({\bf
q})$. We will show ${\bf q}=Q(x)$. This will imply that $t=0$ as $f({\bf q})$
is the first positive return time to $\mathcal{G}$ (see Remark 4.7) and $0\leq
t<f({\bf q})$. This will also imply, as desired, that ${\bf q}=Q(x)=Q(z)$. To
show ${\bf q}=Q(x)$, pick $x_{n}\in\mathcal{V}$ such that $Q(x_{n})\to{\bf q}$
as $n\to\infty$. As $P$ is continuous, we see that $x_{n}=P(Q(x_{n}))\to
P({\bf q})=x$ as $n\to\infty$. Let ${\bf t}={\bf t}(x,Q(x))$. For $N>0$, let

$E_{N}=\bigcup_{i=-N}^{N-1}[t_{i}+\beta/2,t_{i+1}-\beta/2]$

and define the open set

$O_{N}=\left(\bigcap_{i=-N}^{N}\Phi_{-(t_{i}-\beta/2,t_{i}+\beta/2)}\text{\rm
Int}^{\Phi}(S_{Q_{i}(x)})\right)\setminus\Phi_{-E_{N}}\mathcal{G}.$

Note that for $i=-N,\ldots,N$, $\Phi_{(t_{i}-\beta/2,t_{i}+\beta/2)}x\in
Q_{i}(x)$ and $x\notin\Phi_{[t_{i}+\beta/2,t_{i+1}-\beta/2]}\mathcal{G}$. As
$x\in\mathcal{V}$, it holds that for all $t\in\mathbb{R}$,
$\Phi_{t}x\notin\mathcal{Z}$. Thus, if $\Phi_{t}x\in S_{i}$, then in effect
$\Phi_{t}x\in\text{\rm Int}^{\Phi}S_{i}$. We conclude $x\in O_{N}$ is open and
contains $x$.

Note that if $y\in O_{N}\cap\mathcal{V}$, then $Q_{i}(y)=Q_{i}(x)$ for $-N\leq
i\leq N$. It follows that for each $N$, there exists $m(N)$ such that for
$n>m(N)$, $Q_{i}(x_{n})=Q_{i}(x)$ for $-N\leq i\leq N$. This implies that
$Q(x_{n})\to Q(x)$ as $n\to\infty$. Recall that $Q(x_{n})\to{\bf q}$ as
$n\to\infty$. Thus we obtain that ${\bf q}=Q(x)$ as desired. ∎

###  4.3. Bowen and Walters’ question answered

Bowen and Walters asked the following question:

######  Question.

([BW72, Problem, p. 192]) Let $(X,\Phi)$ be an expansive topological flow
without fixed points. Can the symbolic extension $\pi$ in Theorem 4.9 be made
entropy preserving by choosing carefully the cross-sections
$\\{S_{i}\\}_{i=1}^{N}$?

######  Theorem 4.11.

Let $(X,\Phi)$ be an expansive flow without fixed points. Then $\pi$ is
strongly isomorphic.

###### Proof.

By [KS81, Theorem 4.2] an expansive flow is finite-dimensional. By [BW72,
Theorem 5], an expansive flow has the property that for any $\tau>0$, the
number of periodic orbits of period less than $\tau$ is finite. Since
$(X,\Phi)$ has no fixed points, we can now apply Main Theorem 3.14 to conclude
$(X,\Phi)$ has the small flow boundary property. Adapting the notation and
results of Subsection 4.2, we notice that the cross-sections $S_{i}$ in Lemma
5.1 may be chosen such that the $S_{i}$ have small flow boundaries for each
$i$. Thus $\Phi_{[-\eta,\eta]}(\mathcal{Z})$ is a null set and by
$\sigma$-additivity of measures, $\mathcal{W}$ defined in equation 11 is a
full set. It therefore suffices to show that $\pi$ is one-to-one when
restricted to $\pi^{-1}(\mathcal{W})$. In other words, if $\pi(({\bf
q_{1}},t_{1}))=\pi(({\bf q_{2}},t_{2}))=y\in\mathcal{W}$, then $({\bf
q_{1}},t_{1})=({\bf q_{2}},t_{2})$. Write $y=\Phi_{t}x$ for some
$t\in\mathbb{R}$ and $x\in\mathcal{V}$. By applying $\Phi_{-t}$, we may assume
w.l.o.g. $\pi(({\bf q_{1}},t_{1}))=\pi(({\bf q_{2}},t_{2}))\in\mathcal{V}$. It
is thus enough to show that $\pi$ is one-to-one when restricted to
$\pi^{-1}(\mathcal{V})$. This follows from Lemma 4.10. ∎

The following theorem gives a strong positive answer to Bowen and Walters’
question above.

######  Theorem 4.12.

(=Theorem B) Let $(X,\Phi)$ be an expansive flow. Then it has a strongly
isomorphic symbolic extension.

###### Proof.

If $(X,\Phi)$ has no fixed points then the result follows from Theorem 4.11.
If $(X,\Phi)$ has fixed points then they are isolated ([BW72, Lemma 1]), so
the result follows from the previous case. ∎

##  5\. Appendix

###  5.1. Existence of a complete family

The following lemma is obtained by a slight modification of the proof of Lemma
$7$ of [BW72]. See also [KS81, Lemma 2.4] for a similar construction.

######  Lemma 5.1.

Let $(X,\Phi)$ be a topological flow without fixed points. There is an
$\eta>0$ so that the following holds. For each $\alpha>0$, $z\in X$ and cross-
section $S$ with $z\in\text{\rm Int}^{\Phi}(S)$, there are two finite families
$\mathcal{S}=\\{S_{i}\\}_{i=1}^{N}$ and
$\mathcal{S}^{\prime}=\\{S_{i}^{\prime}\\}_{i=1}^{N}$ of pairwise disjoint
closed cross-sections of injectivity time $\eta$ and diameter at most $\alpha$
so that

  * •

$z\in\text{\rm Int}^{\Phi}(S_{1})\subset S^{\prime}_{1}\subset S$,

  * •

$\overline{S_{i}}\subset\text{\rm Int}^{\Phi}(S_{i}^{\prime})$ for all $1\leq
i\leq N$;

  * •

$\Phi_{[0,\alpha]}\mathcal{G}=\Phi_{[-\alpha,0]}\mathcal{G}=X$,

where $\mathcal{G}=\cup_{1\leq i\leq N}S_{i}$.

###### Proof.

By Theorem 2.8, for each $x\in X$ there is a cross-section $S_{x}$ of
injectivity time $2\eta_{x}>0$ such that $x\in\text{\rm Int}^{\Phi}S_{x}$. By
compactness of $X$, there are $x_{i}\in X$ $(1\leq i\leq n)$ with $x_{1}=z$
and $S_{x_{1}}=S$ such that

$X=\cup_{i=1}^{n}\Phi_{(-\eta_{x_{i}},\eta_{x_{i}})}\text{\rm
Int}^{\Phi}S_{x_{i}}.$

Let $\eta=\min_{1\leq i\leq n}\\{\eta_{x_{i}}\\}$. Then for each $x$ there is
an $x_{i}$ and an $\rho_{x}\in(-\eta_{x_{i}},\eta_{x_{i}})$ with
$x\in\Phi_{\rho_{x}}\text{\rm Int}^{\Phi}S_{x_{i}}$. Let
$T_{x}:=\Phi_{\rho_{x}}S_{x_{i}}$ which is a cross-section of injectivity time
at least $\eta_{x_{i}}\geq 2\eta$ and $x\in\text{\rm Int}^{\Phi}T_{x}$.

Let $\alpha>0$ be given. Choose $\epsilon>0$ sufficiently small such that
$\epsilon\leq\min\\{\alpha/4,\eta\\}$ and
$\operatorname{diam}(\Phi_{r}(A))\leq\alpha$ whenever $|r|\leq\epsilon$ and
$A\subset X$ with $\operatorname{diam}(A)<\epsilon$. For each $x\in X$, let
$V_{x},V_{x}^{\prime}\subset\text{\rm Int}^{\Phi}T_{x}$ be closed
neighborhoods of $x$ in $T_{x}$ with $V_{x}\subset\text{\rm
Int}^{\Phi}V_{x}^{\prime}$ and $\operatorname{diam}(V_{x}^{\prime})<\epsilon$.
Then $x\in\text{\rm Int}^{\Phi}V_{x}\subset V_{x}^{\prime}$ and
$V_{x},V_{x}^{\prime}$ are cross-sections of injectivity time $2\eta$ and
diameter at most $\alpha$. As $X$ is compact, we can find $y_{i}\in X,1\leq
i\leq k$ with $y_{1}=z$ such that

$X=\cup_{i=1}^{k}\Phi_{(-\epsilon,\epsilon)}\text{\rm Int}^{\Phi}V_{y_{i}}.$

We construct finite pairwise disjoint families $\mathcal{S}_{i}$ and
$\mathcal{S}_{i}^{\prime}$ of closed cross-sections recursively. Let
$\mathcal{S}_{1}=\\{V_{y_{1}}\\}$ and
$\mathcal{S}_{1}^{\prime}=\\{V_{y_{1}}^{\prime}\\}$. Suppose
$\mathcal{S}_{i-1}$ and $\mathcal{S}_{i-1}^{\prime}$ have been defined. For
each $y\in V_{y_{i}}^{\prime}$,
$\Phi_{[-\epsilon,\epsilon]}(y)\cap\cup_{S^{\prime}\in\mathcal{S}^{\prime}_{i-1}}S^{\prime}$
is a finite set of points since $\mathcal{S}^{\prime}_{i-1}$ consists of
cross-sections. As $\Phi$ is continuous and
$\cup_{S^{\prime}\in\mathcal{S}^{\prime}_{i-1}}S^{\prime}$ is closed, there
exists a non-empty open interval $I_{y}\subset(-\epsilon,\epsilon)$ and closed
neighborhoods $W_{y},W_{y}^{\prime}$ of $y$ in $V_{y_{i}}$ and
$V_{y_{i}}^{\prime}$ respectively with $W_{y}\subset\text{\rm
Int}^{\Phi}W_{y}^{\prime}$ such that

(12)
$\Phi_{I_{y}}(W_{y}^{\prime})\cap\cup_{S^{\prime}\in\mathcal{S}^{\prime}_{i-1}}S^{\prime}=\emptyset$

Let $y^{1},y^{2},\dots,y^{\ell}$ be points in $V_{y_{i}}^{\prime}$ such that
$W_{y^{1}},\dots,W_{y^{\ell}}$ cover $V_{y_{i}}$ and
$W_{y^{1}}^{\prime},\dots,W_{y^{\ell}}^{\prime}$ cover $V_{y_{i}}^{\prime}$.
Pick distinct $\rho_{1}\in I_{y^{1}},\dots,\rho_{\ell}\in I_{y^{\ell}}$ and
set

$\mathcal{S}_{i}=\mathcal{S}_{i-1}\cup\\{\Phi_{\rho_{1}}(W_{y^{1}}),\dots,\Phi_{\rho_{\ell}}(W_{y^{\ell}})\\}$

and

$\mathcal{S}_{i}^{\prime}=\mathcal{S}_{i-1}^{\prime}\cup\\{\Phi_{\rho_{1}}(W_{y^{1}}^{\prime}),\dots,\Phi_{\rho_{\ell}}(W_{y^{\ell}}^{\prime})\\}.$

By Equation (12) and as $\rho_{1},\dots,\rho_{\ell}$ are distinct, the members
of $\mathcal{S}_{i}$ are pairwise disjoint. The same holds for
$\mathcal{S}^{\prime}_{i}$. Set $\mathcal{S}=\mathcal{S}_{k}$ and
$\mathcal{S}^{\prime}=\mathcal{S}_{k}^{\prime}$. Clearly
$X=\Phi_{[-2\epsilon,2\epsilon]}\mathcal{G}$. For every $x\in X$,
$\Phi_{2\epsilon}x\in\Phi_{[-2\epsilon,2\epsilon]}\mathcal{G}$. Thus
$x\in\Phi_{[-4\epsilon,0]}\mathcal{G}\subset\Phi_{[-\alpha,0]}\mathcal{G}$.
Thus $X=\Phi_{[-\alpha,0]}\mathcal{G}$. Similarly
$X=\Phi_{[0,\alpha]}\mathcal{G}$. Q.E.D. ∎

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