Strongly isomorphic symbolic extensions for expansive topological flows

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.


Introduction
Structure of the paper Acknowledgements 2. Preliminaries 2.1.Notation 2.2.Discrete dynamical systems and topological flows 2.3.Cross-sections 2.4.Flow boundaries and interiors 2.5.Good cross-sections exist 2.6.Calculating with flow interiors and boundaries 2.7.The small flow boundary property 2.8.Suspension flows and symbolic extensions 2.9.Some facts from dimension theory 3. Establishing the small flow boundary property 3.1.Notation and assumptions in force in Section 3 3.2.Local homeomorphisms between cross-sections 3.3.Return time sets and cross-section names 3.4.Statement of the Main Theorem 3.5.General position 3.6.Key Lemma 3.7.Proof of the Main Theorem 1. Introduction Symbolic coding of dynamical systems (X, T ) in the form of measurable factor maps into shift spaces over finite alphabets (X, T ) → (Y, σ) ⊂ ({1, 2, . . ., a} Z , σ) 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, σ) ⊂ ({1, 2, . . ., a} Z , σ) for which the original system occurs as a continuous factor (Y, σ) → (X, T ).In other words this form of "digitization" corresponds to symbolic extensions.However as (Y, σ) is an extension of (X, T ) it is necessarily "more complex" than (X, T ).Viewing (Y, σ) 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 isomorphic 1 .The associated theory for Z-systems is deep and extensive ([BD04, Dow05,Dow11]).Recently a symbolic extension theory for 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 Z-system (X, T ) was introduced by Shub and Weiss in [SW91] who investigated the question under which conditions a given 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 Z-system which has the small boundary property must have mean dimension zero.Moreover a 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 Z-system without periodic points has the small boundary property 2 .
Burguet [Bur19] introduced the small flow boundary property for topological flows as an analog to the small boundary property.He showed that 1 See Definition 2.23 2 More generally, from [Lin95, Theorem 3.3] it follows that an infinite Z-system (|X| = ∞) with a finite number of periodic points has the small boundary property.
flows with the small flow boundary property admit strongly isomorphic zerodimensional extensions and gave necessary and sufficient conditions for the existence of symbolic extensions3 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 points4 and such that for any τ > 0, the number of periodic orbits of period less than τ 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 Φ be a topological flow on X without fixed points, having a countable number of periodic orbits.Then (X, Φ) has the small flow boundary property.
Expansive Z-systems were introduced as early as 1950 by [Utz50].In [Red68] and [KR69] it was shown that expansive Z-systems admit symbolic extensions.From the work of Boyle and Downarowicz [BD04] it follows that an expansive 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, that the topological entropy of an expansive flow is finite and that all fixed points of an expansive flow are isolated.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, Φ) 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]).

Preliminaries
If S is clear from the context, it may be omitted from the notation.Let C = {A 1 , . . .A n } be a set of sets A i ⊂ X, i = 1, . . ., n.We denote by Denote by ∂ S V and Int S V respectively the boundary and interior of V w.r.t. the subspace topology induced by S. Let S ⊂ X be a closed subset and 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 → X is a homeomorphism.For Y , a second countable metrizable space we denote by dim(Y ) the Lebesgue covering dimension of Y . 5We use the convention dim(Y ) = −1 iff Y = ∅.Definition 2.1.A topological flow is a pair (X, Φ) where X is a compact metrizable space and Φ : X × R → X is a continuous flow on X, that is, the map Φ is continuous, Φ(•, 0) is the identity map on X and Φ(Φ(x, t), s) = Φ(x, t + s) for all t, s ∈ R and x ∈ X.For t ∈ R, we sometimes use the notation Φ t (x) = Φ(x, t) and notice that Φ t : X → X is a homeomorphism.In addition for Throughout the text, we fix a compatible metric d on X.Given a set x and Φ t (x) = x for any 0 < t < τ .In the latter case, the set {Φ t (x) | 0 ≤ t ≤ τ } is called the periodic orbit associated with x.
Definition 2.3.The flow (X, Φ) is said to be aperiodic if it has no periodic orbits, that is, the equation Φ t (x) = x implies t = 0. 2.3.Cross-sections.Definition 2.4.A cross-section of injectivity time η > 0 is a subset S ⊂ X such that the restriction of Φ on S × [−η, η] is one-to-one.The crosssection is said to be global if there is ξ > 0 such that Φ(S × [−ξ, ξ]) = X.The set Φ [−η,η] (S) is called the η-cylinder associated with S.
Remark 2.5.In [Bur19] a closed cross-section S such that the flow map Φ : (x, t) → Φ t (x) is a surjective local homeomorphism from S × 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, Φ) be a topological flow.A finite family A trivial observation is if x ∈ S and Φ t x ∈ S for some t = 0, where S is a cross-section of injectivity time η > 0, then |t| > 2η as otherwise Definition 2.7.Let U be a set contained in a closed cross-section of injectivity time η.The flow interior Int Φ (U ) ⊂ U of U is the unique set obeying: The flow boundary ∂ Φ U of U is the unique set obeying: One can show that for every 0 < γ < η: Note that ∂ Φ U is closed as it can be written as ) for any 0 < γ < η (recall that U is the closure of U w.r.t. the space X).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, Φ) be a topological flow without fixed points, then for each x ∈ X there is a closed cross-section S such that x ∈ Int Φ S.
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, Φ) be a topological flow with dim(X) = n ≥ 1.Let S ⊂ X be a closed cross-section.Then dim(S) ≤ n − 1.If in addition S is a global cross-section then dim(S) = n − 1.
Proof.As S × [−η, η] is homeomorphic to a subset of X, one has dim(S × [−η, η]) ≤ n.By a theorem of Hurewicz in [Hur35], as S is compact, dim(S) ≤ n − 1.Now assume in addition that S is a global cross-section.Let ξ > 0 such that the natural continuous map f : S × [−ξ, ξ] → X given by (x, t) → Φ(x, t) is surjective.Note that f is a closed map between two metric separable spaces and for every x ∈ X, f −1 (x) is countable.Thus by [Eng95, Theorem 1.12.4] where we used the product theorem for non-empty metric separable spaces A and B dim(A × B) ≤ dim(A) + dim(B) ([Eng95, Theorem 1.5.16]).Thus dim(S) ≥ n − 1 as desired.
We remark that one can not replace the assumption "global cross-section S" by "a cross-section with Int Φ (S) = ∅".A counter-example is that X is a disjoint union of X 1 and X 2 with dim(X 1 ) > dim(X 2 ), Φ is a continuous flow on X without fixed points and S ⊂ X 2 is a cross-section with Int Φ (S) = ∅.Then it is clear that dim(S) ≤ dim(X 2 ) − 1 < dim(X) − 1.
2.6.Calculating with flow interiors and boundaries.Let S, V ⊂ X. Recall that ∂ S V and 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 ⊂ S. Then Example 2.12.Consider a minimal rotation on the torus (T 2 , Φ). Identify . 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 ⊂ S.
Proof.By Lemma 2.11 and Definition 2.7 it follows on the one hand that Lemma 2.14.Let (X, Φ) be a topological flow.Let U ⊂ S be two subsets of a closed cross-section.Then . This expression equals the empty set as by assumption Given a cross-section S and x ∈ Int Φ S, we clearly have δ := dist(x, ∂ Φ S) > 0 as ∂ Φ S is closed.Therefore it is easy to see that U = B(x, δ/2) ∩ S is a relatively open set in S with x ∈ U ⊂ Int Φ S and U ∩ ∂ Φ S = ∅.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, Φ) be a topological flow.Let U be a subset of a closed cross-section S such that U ∩ ∂ Φ S = ∅.Then: Remark 2.16.A case where the previous remark can be used is when S is a cross-section, U, V ⊂ S with Int Φ U = U and Int Φ V = V and 2.7.The small flow boundary property.First, we recall the small boundary property for discrete dynamical system.Definition 2.17.([SW91, LW00]) Let (X, T ) a discrete dynamical system.A subset A ⊂ X has a small boundary if µ(∂A) = 0 for every T -invariant measure µ.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, Φ) 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 Φ-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 η has a small flow boundary For a cross-section A ⊂ X, we define the counting orbit capacity of A by ocap ♯ (A) := lim The limit exists and is finite as Lemma 2.20 ([Bur19], Lemma 2.10).Let (X, Φ) be a topological flow.Suppose that S is a closed cross-section of injectivity time η > 0. Then the following are equivalent.
Definition 2.21.Let (X, Φ) be a topological flow.The flow (X, Φ) is said to have the small flow boundary property if for any x ∈ X and any closed cross-section S ′ with x ∈ Int Φ (S ′ ), there exists a closed subset S ⊂ S ′ such that x ∈ Int Φ (S) and S has a small flow boundary.
Remark 2.22.Note that if a topological flow has the small flow boundary property than it has no fixed points as it is required that every x ∈ X belongs to a cross-section.
2.8.Suspension flows and symbolic extensions.Let (Z, ρ) be a compact metric space and The suspension flow of T under f is the flow Φ on the space A suspension flow over a zero-dimensional 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 Z-topological dynamical system (a.k.a 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 2.23.Let (X, Φ) and (Y, Ψ) be two topological flows.Suppose that π : Y → X is a topological extension from (X, Φ) to (Y, Ψ).A topological extension is said to be (see [Bur19, §2.3]) • entropy-preserving when it preserves topological entropy i.e. h top (X, Φ) = h top (Y, Ψ). • principal when it preserves the entropy of invariant measures, i.e. h(µ, Ψ) = h(π * µ, Φ) for all Ψ-invariant measures µ, Remark 2.24.Clearly, strongly isomorphic =⇒ isomorphic =⇒ principal=⇒ entropy-preserving.It is easy to give an example of an extension which is entropy-preserving but not principal: An example of an extension which is principal but not isomorphic is given by [BD19, Theorem 4.7].Downarowicz and Glasner constructed an minimal system which is an isomorphic but not almost 1-1 extension of its maximal equicontinuous factor.If this extension were strongly isomorphic then it would trivially have at least one singleton fiber implying the extension is almost 1-1.Thus their construction also gives an example of an extension which is isomorphic but not strongly isomorphic (see [DG16, Remark 3.2(d)]).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 Θ : M(X, T ) → M(Z 1 (X), Φ) given by µ → µ × λ, where λ 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 σ set is a countable union of closed sets.
Theorem 2.25.([Eng95, Proposition 1.2.12])Let E be a zero-dimensional subset in a separable metric space M .Then for every x ∈ M and every open neighborhood The following corollary is obvious.
Corollary 2.27.Let (B i ) i∈N be a countable collection of F σ sets in a separable metric space satisfying dim B i ≤ k for all i.Then dim i B i ≤ k.
Lemma 2.28.Let M be a separable metric space.Let C be closed and U open then Proof.Note M is second-countable and write U as a countable union of closed balls.
The following theorem is well known.
We will need the following strengthening of Theorem 2.29.
Theorem 2.30.Let M be a separable metric space.
i=1 be a countable basis of M so that dim(∂B i ) ≤ n − 1 for all i.Using the inductive assumption let F i ⊂ ∂B i be a zero-dimensional As an illustration of Theorem 2.30 consider M = [0, 1] 2 and note that E = M ∩(Q×Q) is zero-dimensional and M \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 \ E. As an illustration of Theorem 2.25 note that balls centered at rational coordinates with a transcendental radius do not intersect E.
Proposition 2.31.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 3. Establishing the small flow boundary property 3.1.Notation and assumptions in force in Section 3. Let (X, Φ) be a topological flow without fixed points.Through this section, we always suppose that dim(X) = d + 1 for d ≥ 0. By Lemma 5.1 there exists an η > 0 and 0 < α < η such that (X, Φ) has two complete families of cross-sections S = {S i } N i=1 of (closed disjoint) cross-sections of injectivity time η S and Starting with Definition 3.18 in the sequel we fix two cross-sections S = S i and S ′ = S ′ i in the complete families S and S ′ for some 1 ≤ i ≤ N .

Local homeomorphisms between cross-sections.
Definition 3.1.Let 1 ≤ i, j ≤ N .Denote by t i,j the first positive hitting time from S i to S j , that is, for x ∈ S i , where if the argument set is empty we put t i,j (x) = ∞.The functions t ′ i,j are defined similarly w.r.t.S ′ i and S ′ j .Note t ′ i,j (x) ≤ t i,j (x) for all x ∈ S i .Define D i,j = {x ∈ S i : t i,j (x) ≤ η} and D ′ i,j = {x ∈ S ′ i : t ′ i,j (x) < ∞}.Note D i,i = ∅ for all i.Let T i,j be the first positive hitting map from D i,j to S j , that is, for x ∈ D i,j , Similarly, we define T ′ i,j from D ′ i,j to S ′ j .Definition 3.2.Define the first return-time map t (1) i=1 is a complete family of closed disjoint cross-sections, there is γ > 0 so that S = N i=1 S i is a global cross-section with injectivity time γ.Thus the argument set in the definition above is never empty.
Inductively for i ≥ 1 define the (i + 1)-st return time map t That is F i,j is the set of x ∈ S i such that the first member of S it hits after leaving S i is S j .Lemma 3.4.If x ∈ F i,j , then t i,j (x) > γ.
Proof.Note t G is bounded from below by 2γ.
. Since Φ is continuous, there is an δ > 0 such that for any y ∈ X with dist(x, y) < δ, we have that Φ(y, i ) > 0, thus taking δ small enough, one may assume w.l.o.g. that whenever y ∈ V δ .In particular t ′ i,j (y) < 2η for all y ∈ V δ and T ′ i,j (V ) ⊂ Int Φ S ′ j .We claim that after taking δ small enough, the map t ′ i,j is continuous on V = V δ .If not, then there exists a sequence {y n } n∈N ⊂ V such that lim n→∞ . As by definition 0 < t i,j (x) ≤ η and by choice ǫ < η, one has that t x ≤ t i,j (x) + ǫ < 2η and Φ(x, this is a contradiction.Let z = T ′ i,j (x ′ ) ∈ Int Φ S ′ j for some x ′ ∈ V .Let W ρ = {y ∈ S ′ j : dist(z, y) < ρ} be an open neighborhood of z in S ′ j .Using Equation (1) with y = x ′ , for ρ > 0 small enough, there is 0 < ξ < η such that Φ −t i,j (x) (Φ (−ρ,ρ) W ρ ) is an open set in Φ (−ξ,ξ) (V ).Thus for each w ∈ W ρ , there are unique v ∈ V , r ∈ (−ξ, ξ), such that Φ t i,j (x)−r (v) = w.By Lemma 3.6, one must have t ′ i,j (v) = t i,j (x) − r and Thus in order to establish that

3.3.
Return time sets and cross-section names.
Proposition 3.8.For every 1 ≤ i, j ≤ N there is a finite collection of open sets C i,j in S ′ i with the following properties: ⊂ D i,j so that x q → q→∞ x.As t i,j (x q ) ≤ η, we may assume w.l.o.g.t i,j (x q ) converges to some t ≤ η.Clearly Φ t (x) ∈ S j and therefore x ∈ D i,j .Conclude F i,j ⊂ D i,j ⊂ S i ⊂ Int Φ S ′ i .Using Proposition 3.7 cover F i,j by a finite collection of open sets C i,j such that C i,j ⊂ C i,j ⊂ Int Φ S ′ i .with properties (2) and (3).
For each C ∈ C i,j , we define on the set 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 ∈ N. Let V ⊂ X, not necessarily contained in the image of T C n .Following standard notation, we denote Definition 3.9.Let n ∈ N ∪ {0}.Define the return time set: Then the following holds: (1) The first claim is best understood by considering the first cases.Indeed note For Claim (2), note Using Proposition 3.8 let δ 0 > 0 small enough so that 2δ 0 is a Lebesgue number for each of the open covers C i,j (of F i,j ).Thus for all 1 ≤ i, j ≤ N , (3) From now on in this section we abbreviate Θ Definition 3.12.We define Recall that dim(X) = d + 1.The following lemma will be important in the sequel: Lemma 3.13.Let x ∈ S 1 so that there are 0 = t 0 < t 1 < t 2 < . . .< t d with Φ t k x ∈ S 1 for k = 0, . . ., d.Then there exists C n = (C 0 , C 1 , . . ., C n−1 , C n ) ∈ C(n) for some n ∈ N, and j 0 = 0 < j 1 < . . .< j d such that x ∈ Z δ 0 C n and T j i C n x = Φ t i x for i = 0, . . ., d.

Statement of the Main Theorem.
The main result is as follows.
Theorem 3.14 (=Theorem A).Let X be a finite-dimensional space.Let Φ be a topological flow on X without fixed points, having a countable number of periodic orbits.Then (X, Φ) 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 Per(X), for every pair of open sets U, V ⊂ X, such that ∂U \ Per(X) ⊂ V , there is an open set U ′ with U ⊂ U ′ ⊂ U ∪ V , such that ∂U ′ is the union of a set of zero (discrete) orbit capacity and a subset of Per(X).This statement should be compared with Theorem 3.26.Our proof is not a routine generalization.Using the same method, it is very probable one could prove an analogous 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 τ > 0 the number of periodic orbits of period less than τ is finite, has the small flow boundary property.A key tool in Burguet's proof is what he calls the n-transverse property (≈ 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 M1 and M2 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 ∈ Int Φ S, where S is a closed cross-section.We may find U, V ⊂ S with x ∈ Actually we establish the stronger property that for every y ∈ S, ).Consider a small neighborhood of x ∈ ∂ Φ U ′ and consider the return profile of this neighborhood: i.e., how elements in this neighborhood return to ∂ Φ U ′ .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.
Remark 3.15.From the proof of Theorem 3.14 (see the beginning of the proof of Lemma 3.21), it follows that one may replace the assumption of the countability of periodic orbits by the assumption that the periodic orbits intersect any cross-section in a zero dimension set, in the statement of the theorem.
3.5.General position.By Lemma 2.10, a global closed cross-section in the (d + 1)-dimensional space X has dimension d.We will therefore use the following definition: Definition 3.16.A collection A of subsets in a d-dimensional space is said to be in general position if for every finite B ⊂ A, one has that 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 d-dimensional space then generically any two will have intersection with dimension less than d − 2, etc. Remark 3.17.For the purposes of the proof the most important consequence of Definition 3.16 is that every finite sub-collection B ⊂ A with d + 1 elements has empty intersection.
In what follows in this section, we fix two cross-sections S = S i and S ′ = S ′ i in the complete families S and S ′ for some 1 ≤ i ≤ N .Recall the definition of T −k C n in Equation (2) and the definition of I C n in Definition 3.9 in Subsection 3.3.
The following remark is easy to establish: Assuming that such U ′ has been constructed, we may complete the proof by inducting on F 1 , finally obtaining F 0 = C(n).Thus it remains to construct U ′ that satisfies the conditions (B1) − (B5) which will be achieved in the sequel.Indeed set U ′ = A m of Lemma 3.24.
We will now construct a ball around x, B x = B(x, r) for some r > 0, such that B x ⊂ V and for any C ∈ C(n) and for any distinct j, ℓ ∈ I C , one has that with j > ℓ (as one may take the intersection of the open balls associated to each C n and each j > ℓ).Assume for a contradiction that for every p ∈ N we may find an open neighborhood W p in S ′ of x with diam(W p ) < 1 p and w p ∈ X such that C n w p , y p := T ℓ C n w p ∈ W p .This implies we have found sequences (y p ) p∈N and (z p ) p∈N in X converging to x such that T j−ℓ D j−ℓ y p = z p , where D j−ℓ = (C ℓ , . . ., C j ) ∈ C(j − ℓ).Thus by Proposition 3.8 there exist t p ≤ 2η(j − ℓ) such that Φ tp y p = z p .W.l.o.g.t p → t, for k → ∞ for some t ≤ 2η(j − ℓ).We conclude that Φ t x = x which is a contradiction to the fact that x / ∈ P as (A5) ∂ Φ A 0 ∩ P = ∅.Let now {B i } m i=1 be a finite subcover of ∂ Φ A 0 extracted from {B x }.Clearly properties (D1)-(D3) hold.Lemma 3.24.Let (B k ) and (B ′ k ) be as in Lemma 3.23.There exists sets Proof.We will construct sets {A k } m k=1 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 ) k−1 i=0 have been already defined, satisfying the listed proprieties.Define: Clearly, E is a zero-dimensional set in S ′ by Corollary 2.27 and Lemma 3.11(2).Note that by the inductive assumption Thus one may find ǫ > 0 so that: where recall from Subsection 2.1 that the closed ǫ-tube around a closed set We will show that A k is the set we look for.By (F1), Conditions (E1) and (E5) hold.Since Int Φ W = W , (E2) follows by Lemma 2.14.Note: 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 ∈ C(n − 1) ∪ F 0 .Assume for a contradiction that A k is not C-general.We can thus find By (D1) and the fact that W ⊂ B ′ k , we see that ∩ j∈I K α j j = ∅ if α j = 1 for at least two j ∈ I.By the induction hypothesis, we have that dim(∩ j∈I K 0 j ) ≤ max{−1, d−♯I}.By Lemmas 2.28 and 3.11 the sets K α j j are F σ .As the intersection of finitely many F σ sets is an F σ set, we may apply the sum theorem for F σ sets (Corollary 2.27).It follows that there is ℓ ∈ I such that A which is a contradiction to (F 3).We conclude By the inductive hypothesis (E6) holds for k − 1 and may be applied on ∩ j∈I\{ℓ} K 0 j .Thus, using Equation (5): This is a contradiction to Equation (7).We now prove (E7).We notice from (F2) and the fact that The (RHS) equals the union of the four following expressions, one of which is empty: as the union of 3 |I| expressions.Notice that as above each one of these expressions is a finite intersection of F σ sets, thus F σ by itself.Therefore by Corollary 2.27, 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 ∩ ∂ Φ W appears twice, then the expression is empty as Thus the expressions involving ∂ Φ W appear at most once.If such expression does not appear at all, then we are only left with the expression k−1 i=1 B i ∩ ∂ Φ 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 ∂ Φ W appears exactly once.We thus analyze an expression contained in an expression of the form which is a contradiction to (F 3).Thus, using condition (E7) for k − 1 and Equation (5), as desired.
3.7.Proof of the Main Theorem.The proof of the main theorem necessitates an approximation lemma.Recall the number δ 0 was defined before Definition 3.12.
Lemma 3.25.Let A and V be subsets such that Proof.We claim there is m 0 ∈ N, so that for all m ≥ m 0 , U = U m := Θ 1/m (A) fulfills the sought after requirements.Assume for a contradiction that this is not the case.We may thus find is finite, by passing to a subsequence we may assume w.l.o.g.C n m = C n for some fixed C n = (C 0 , C 1 , . . ., C n ) ∈ C(n), and j m i = j i for i = 0, . . ., d, for some fixed By passing to a subsequence we may assume w.l.o.g. that x m → x ∈ Z δ 0 C n .In particular T j i C n x is well defined for i = 0, . . ., d.
Theorem 3.26.Let X be a compact finite-dimensional space.Let Φ be a topological flow on X without fixed points, having a countable number of periodic orbits.Let S = {S i } N i=1 be a complete families of cross-sections of (X, Φ).Then for any U, V ⊂ S with Proof.By induction, we will construct {U k } ∞ k=0 and {V k } ∞ k=0 satisfying for all k ≥ 0, (1) . By (4) and (5), we have that for all k, ℓ > 0. It follows that and thus as by (4) for all k > 0. Then by (2), (3), (4) we obtain that (10) for all k > 0. In order to establish ocap ♯ (∂ Φ U ′ ) = 0, it is sufficient to show that any orbit of the flow hits the set ∂ Φ U ′ at most d times.Let x ∈ ∂ Φ U ′ ⊂ S and assume for a contradiction that there are 0 < t 1 < t 2 < . . .< t d such that Φ t i x ∈ ∂ Φ U ′ ⊂ S, i = 1, . . ., d.By Lemma 3.13, for some n, there is which is a contradiction to (6).
growth7 and the topological entropy of an expansive discrete system is finite [Con68].Mañé [Mañ79] showed that an expansive Z-system must be finitedimensional.From this it follows that an expansive Z-system without fixed points has the small boundary property ( [Lin95]).By [Bur19, Proposition C.1] an expansive 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, Φ) is said to be expansive if for any ǫ > 0, there exists δ > 0 such that if dist(Φ t (x), Φ s(t) y) < δ for all t ∈ R, for a pair of points x, y ∈ X and a continuous map8 s : R → R with s(0) = 0, then there exists t ∈ R with |t| < ǫ such that y = Φ 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 ≥ 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 point9 .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, following the exposition in [BW72, Section 5].We start by proving lemmas which do not necessitate assuming expansiveness.
Let (X, Φ) be a topological flow without fixed points.By Lemma 5.1, we may find constants 0 < α < η such that there is a complete family S = {S i } N i=1 of injectivity time η > 0 with diam(S i ) ≤ α for all i and Φ [0,α] G = Φ [−α,0] G = X, where G = ∪ 1≤i≤N S i .Since S i are pairwise disjoint and compact, there is β > 0 so that Φ [0,β] (S i ) are pairwise disjoint for all i.For x ∈ G, let r i (x) be the i-th return time to G (see Definition 3.2).As Φ [−α,0] G = X, every y ∈ G, has some x ∈ G and 0 ≤ s ≤ α so that Φ −s x = y, i.e.Φ s y = x ∈ G.It follows that 0 < β ≤ r i+1 (x) − r i (x) ≤ α for all x ∈ G. Denote by Σ = {1, 2, . . ., N } and σ the (left) shift on Σ Z .Define The sequence t appearing in the definition is called a return-time sequence for (x, q); q is called a return-name sequence for x and x is called a realization of q.A return-time sequence t for x is any return-time sequence of the form (x, q) ∈ G ⋊ Σ Z for some q.
Note that the sequence r = {r Other return-time sequences than r for x may exist.This is the case if for example for some i, r i+1 − r i−1 ≤ α, as one can remove r i from r and still have a return-time sequence for x.In contrast Lemma 4.3 shows that (x, q) has a unique return-time sequence.In addition in Lemma 4.6, assuming that (X, Φ) is expansive, it is shown that the return-name sequence determines a realization uniquely.Lemma 4.2.Let (X, Φ) be a topological flow without fixed points.Then Proof.Let (x n , q n ) ∈ G ⋊ Σ Z which converge to (x, q) as n → ∞.Let t n be a return-time sequence of (x n , q n ).Since |t n i | ≤ |i|α for every i ∈ Z, we may assume that the limit lim n→∞ t n i exists, say t i for each i ∈ Z. Since Σ is finite, we see that for each i ∈ Z, q n i = q i for large n > n(i).Since S q i is closed for each i ∈ Z, we obtain that Φ t i (x) = lim n→∞ Φ t n i (x n ) ∈ S q i for each i ∈ Z.This implies that (x, q) ∈ G ⋊ Σ Z with return-time sequence t = (t i ) i∈Z .
Lemma 4.3.Let (X, Φ) be a topological flow without fixed points.Then for each (x, q) ∈ G ⋊ Σ Z , the return-time sequence t is unique.
Proof.Suppose that t and t ′ are two distinct return-time sequences of (x, q) ∈ G ⋊ Σ Z .Without loss of generality, we assume that (t Thus we may define the map: Lemma 4.4.Let (X, Φ) be a topological flow without fixed points.Then the map t is continuous.P 2 (G ⋊ Σ Z ).Let P 1 : G ⋊ Σ Z → G be the projection on the first coordinate.Define the continuous map P : Λ → G by P (q) = P 1 (P −1 2 (q)) for q ∈ Λ.Thus, P associates with a return-name sequence in Λ its realization in G ⊂ X.Clearly, P • Q is the identity on V.
We define the continuous function f : Λ → [β, α] by f (q) = t 1 (P −1 2 q).Remark 4.7.From the proof of Lemma 4.3 it is clear that f (q) is the first positive time when the realization of q in G returns to S q 1 but it is possible that it returns to G earlier.However, if q ∈ Q(V) then f (q) is the first positive return time to G of the realization of q (due to the definition above of Q involving maximal return time-sequences).
The following lemma is crucial for the proof of Theorem 4.11 in the sequel.

Proof. By definition π(Λ
Moreover by the definition and equivariance of π, x = π(q, 0) = P (q).We will show q = Q(x).Note that as z = Φ t (x), x ∈ V, t is a return-time of x to G. By assumption 0 ≤ t < f (q) and f (q) is the first positive return time to G (see Remark 4.7).Thus q = Q(x) implies that t = 0 and from this it holds as desired, that q and define the open set As x ∈ V, it holds that for all t ∈ R, Φ t x / ∈ Z.Thus, if Φ t x ∈ S i , then in effect Φ t x ∈ Int Φ S i .We conclude O N is open and contains x.
Note that if y ∈ O N ∩ V, then Q i (y) = Q i (x) for −N ≤ i ≤ 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 ≤ i ≤ N .This implies that Q(x n ) → Q(x) as n → ∞.Recall that Q(x n ) → q as n → ∞.Thus we obtain that 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, Φ) be an expansive topological flow without fixed points.Can the symbolic extension π in Theorem 4.9 be made entropy-preserving by choosing carefully the cross-sections {S i } N i=1 ?Theorem 4.11.Let (X, Φ) be an expansive flow without fixed points.Then π 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 τ > 0, the number of periodic orbits of period less than τ is finite.Since (X, Φ) has no fixed points, we can now apply Main Theorem 3.14 to conclude (X, Φ) 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 Φ [−η,η] (Z) is a null set and by σ-additivity of measures, W defined in Equation (11) is a full set.It therefore suffices to show that π is one-to-one when restricted to π −1 (W).In other words, if π((q 1 , t 1 )) = π((q 2 , t 2 )) = y ∈ W, then (q 1 , t 1 ) = (q 2 , t 2 ).Write y = Φ t x for some t ∈ R and x ∈ V = W ∩ G.By applying Φ −t , we may assume w.l.o.g.π((q 1 , t 1 )) = π((q 2 , t 2 )) ∈ V.It is thus enough to show that π is one-to-one when restricted to π −1 (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, Φ) be an expansive flow.Then it has a strongly isomorphic symbolic extension.
Proof.If (X, Φ) has no fixed points then the result follows from Theorem 4.11.If (X, Φ) has fixed points then they are isolated ([BW72, Lemma 1]), so the result follows from the previous case.
Remark 4.13.It is possible to strengthen the above theorem by achieving a strongly isomorphic symbolic extension π : (Λ f , Ψ) → (X, Φ) which is at same time uniformly finite-to-one, that is there exists K > 0 so that for all x ∈ X, |π −1 (x)| ≤ K.The proof will be included in a forthcoming work.

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, Φ) be a topological flow without fixed points.There is an η > 0 so that the following holds.For each α > 0, z ∈ X and crosssection S with z ∈ Int Φ (S), there are two finite families S = {S i } N i=1 and S ′ = {S ′ i } N i=1 of pairwise disjoint closed cross-sections of injectivity time η and diameter at most α so that • z ∈ Int Φ (S 1 ) ⊂ S ′ 1 ⊂ S, where G = ∪ 1≤i≤N S i .Proof.By Theorem 2.8, for each x ∈ X there is a cross-section S x of injectivity time 2η x > 0 such that x ∈ Int Φ S x .By compactness of X, there are x i ∈ X (1 ≤ i ≤ n) with x 1 = z and S x 1 = S such that X = ∪ n i=1 Φ (−ηx i ,ηx i ) Int Φ S x i .Let η = min 1≤i≤n {η x i }.Then for each x there is an x i and an ρ x ∈ (−η x i , η x i ) with x ∈ Φ ρx Int Φ S x i .Let T x := Φ ρx S x i which is a cross-section of injectivity time at least η x i ≥ 2η and x ∈ Int Φ T x .

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].
have a domain which is an an open set in S ′ i 0 and a range which is an open set in S ′ i 1 and S ′ i 2 respectively by Proposition 3.7.Now assume we have proven that the domain ofT k C n is an open set D k in S ′ i 0 and that the range of T k C n is an open set R k in S ′ i k .It is easy to see that the domain of T k+1 C n is an open set in D k and therefore in S ′ i 0 , (T k Cn ) −1 (R k ∩ C k ) and that the range of T k+1 C n is an open set in S ′ i k+1 , T C k (R k ∩ C k ).Note that Z C n equalsthe domain of T n C n which we have seen to be open.Using Proposition 3.7(3), we have that T k C n (Z C n ) is open.By the above T k C n is a homeomorphism on an open domain which contains Z C n and therefore is a homeomorphism on Z C n .