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Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces

We introduce topological definitions of expansivity, shadowing, and chain recurrence for homeomorphisms. They generalize the usual definitions for metric spaces. We prove various theorems about topologically Anosov homeomorphisms (maps that are expansive and have the shadowing property) on noncompact and non-metrizable spaces that generalize theorems for such homeomorphisms on compact metric spaces. The main result is a generalization of Smale's spectral decomposition theorem to topologically Anosov homeomorphisms on first countable locally compact paracompact Hausdorff spaces.


Introduction
The goal of this paper is to extend the following result, Smale's spectral decomposition theorem applied to Anosov diffeomorphisms of compact manifolds, to more general topological spaces.
Theorem 1 ( [15]). Let M be a compact manifold and f : M → M be an Anosov diffeomorphism. Then the non-wandering set Ω(f ) can be written as a finite union of disjoint closed invariant sets on which f is topologically transitive.
Recall that a diffeomorphism is Anosov if it has a hyperbolic structure on the entire manifold and that every Anosov diffeomorphism is expansive and has the shadowing property (see, e.g., [14,Sect. 9.2]). Theorem 1 has been extended to homeomorphisms on compact metric spaces: Yang extended Theorem 2 to noncompact metric spaces ( [18,Theorem 4]), with the additional (strong) requirement that the chain recurrent set be compact.
It is well known that some dynamical properties of homeomorphisms on compact spaces (i.e., properties that are conjugacy invariant), may not be dynamical properties on noncompact spaces. For example, a dynamical system on a noncompact metric space may be expansive or have the shadowing property with respect to one metric, but not with respect to another metric that induces the same topology (see Examples 6 and 7). Of course, it is preferable to have a theory that is independent of any change of (compatible) metric. In this article we give topological definitions of expansiveness and shadowing that are equivalent to the usual metric definitions for homeomorphisms on compact metric spaces and are dynamical properties for any metric space. Moreover, these definitions apply to non-metrizable spaces as well. For related work see [3,6,7,8,11,12,13].
Then we extend the spectral decomposition theorem to dynamical systems on spaces that are not necessarily metrizable and not necessarily compact. The only concession is that in the noncompact case, the collection of basic sets in the decomposition need not be finite.
Theorem 3. Let X be a first countable locally compact paracompact Hausdorff space and f : X → X be an expansive homeomorphism with the shadowing property.
Then Ω(f ) can be written as a union of disjoint closed invariant sets on which f is topologically transitive. If X is compact, then this decomposition is finite.
The paper is organized as follows. We discuss definitions and preliminaries in the next section. In Section 3, we show that topologically Anosov homeomorphisms have a local product structure, and obtain topological stability as a consequence. In Section 4 we investigate the properties of the non-wandering and chain recurrent sets, and prove our main result, Theorem 3. Finally, we discuss the relationships between the new topological definitions of expansiveness and shadowing and the existing metric definitions.

Preliminaries
The following are the standard definitions for expansiveness and shadowing, which we will refer to as metric expansiveness and metric shadowing.
Definition 5. Let (X, d) be a metric space and f : X → X be a homeomorphism.
(1) f is metric expansive if there is an e > 0 such that for any distinct x, y ∈ X, there exists n ∈ Z such that d(f n (x), f n (y)) > e. The number e is called an expansive constant.
These definitions clearly require that X be a metric space. Furthermore, as we see in the following two examples, if X is noncompact, these properties depend on the choice of metric; a homeomorphism that is metric expansive for d may not be for d ′ , even if d and d ′ induce the same topology. Thus on noncompact spaces, these properties are not invariant under topological conjugacy. In addition, if f has these properties, its iterates may not ([4, Example 1], [13,Example 9]). Example 6. Let T : R 2 → R 2 be the linear automorphism induced by the matrix 2 0 0 1 2 , and consider the stereographic projection P : Then T is not expansive if R 2 has the metric induced by P , but T is expansive when R 2 has the usual metric. Moreover, both metrics induce the same topology on R 2 .
with the metric inherited from R 2 , and define f : X → X by f (n, y) = (n + 1, 2y), if n < 0, (n + 1, 1 2 y), if n ≥ 0. Choose δ > 0. Since diam(X n ) < δ for n with |n| > n 0 = ⌈ 1 δ ⌉, the uniform continuity of f and f −1 on the compact set X −n 0 ∪ · · · ∪ X n 0 shows that f has the metric shadowing property. However, f is topologically conjugate to the map on the space n∈Z ({n} × [0, 1]) given by translation on the first coordinate and identity on the second, which clearly does not have the metric shadowing property.
To address these issues, we make purely topological definitions for these notions. Our definitions are equivalent to the metric ones in the case that X is a compact metric space; we discuss the relationships among the definitions in general in Section 5.
Recently, the third and fourth authors gave two topological generalizations of positive expansivity ( [13]). For a given dynamical system f : X → X, they use the product map F ≡ f × f on X × X and neighborhoods of the diagonal ∆ X = {(x, x) : x ∈ X} to extend the notion of positive expansivity. The idea is that x is close to y in X if and only if the point (x, y) is close to the diagonal ∆ X in X × X. Thus, instead of requiring that d(x, y) be less than ε, we can require that (x, y) be in a given neighborhood of ∆ X . This approach is useful for extending dynamical properties from the compact setting to the noncompact setting.
We make the standing assumption that all topological spaces are first countable, locally compact, paracompact, and Hausdorff and that all maps are homeomorphisms. To avoid confusion, we denote subsets of the product space X × X by A, M, U, etc., and subsets of the base space X byȦ,Ṁ ,U , etc.
Let U be a neighborhood of ∆ X , and let U[x] = {y ∈ X : (x, y) ∈ U} be the cross section of U at x ∈ X. For any point x ∈ X and any neighborhoodĠ of x, we can find a neighborhood U of ∆ X such that Note that if M is a neighborhood of ∆ X , then M ∩ M T is a symmetric neighborhood of ∆ X ; thus we can often work with symmetric neighborhoods without loss of generality. Let U n ={(x, y) : there exists z 0 = x, z 1 , . . . , z n = y ∈ X such that (z i−1 , z i ) ∈ U for i = 1, . . . , n}.
The intuition behind this definition is that if we wanted U to be the topological equivalent of having an (ε/n)-ball at each point, then U n would take the place of an ε-ball at each point. Observe that if U is symmetric, then the order of the elements within each ordered pair does not matter. Also, for any neighborhood U of ∆ X we can find a neighborhood V of ∆ X such that F (V 2 ) ⊂ U.
Definition 8. A homeomorphism f : X → X is (topologically) expansive if there is a closed neighborhood N of ∆ X such that for any distinct x, y ∈ X there exists n ∈ Z such that F n (x, y) ∈ N. Such a neighborhood N is called an expansive neighborhood for f . Let E(X) denote the set of expansive homeomorphisms.
Lemma 9. If f is expansive, then it has a proper expansive neighborhood.
Proof. Let N be an expansive neighborhood for f . Since X is locally compact, each x ∈ X has an open, relatively compact neighborhoodU x . Since X is paracompact, the open cover {U x } has a closed (and hence compact) locally finite refinement {V α } ([9, Chapter 5, Theorem 28]).
Definition 10. A homeomorphism f : X → X has the (topological) shadowing property if for every neighborhood E of ∆ X , we can find a neighborhood D of ∆ X such that every D-pseudo-orbit is E-traced by some point y ∈ X. Let S(X) denote the set of homeomorphisms of X with the shadowing property.
Remark 11. If (X, d) is a compact metric space, then for any neighborhood U of ∆ X , we can find δ > 0 such that On the other hand, every U δ is a neighborhood of ∆ X . Thus, the above definitions coincide with the usual notions of expansivity and shadowing on compact metric spaces. However, the following example shows that this argument does not hold if X is not compact.
Example 12. If X is a noncompact metric space, then a neighborhood U of ∆ X may not contain an open set U δ (defined as above) for any The following properties are easy consequences of the definitions. As noted earlier, they do not hold for the metric definitions in the noncompact case.

Proposition 13.
( Lemma 15. Let f be topologically Anosov. If E is a neighborhood of ∆ X such that E 2 is an expansive neighborhood for f , then there is a neighborhood D of ∆ X such that every D-pseudo-orbit is E-traced by exactly one orbit of X. If the D-pseudo-orbit is periodic, then so is the orbit that E-traces it.
Proof. Let E be a symmetric neighborhood of ∆ X such that E 2 is an expansive neighborhood for f . Because f has the shadowing property, there exists a neighborhood D of ∆ X such that any D-pseudo-orbit is E-traced by some point in X. We must now show uniqueness. Suppose Since E 2 is an expansive neighborhood for f , (x, y) ∈ ∆ X , and hence x = y. Now suppose {y k } is a periodic D-pseudo-orbit with period n and let x be the unique E-tracing point. Then {y k+n } is a periodic D-pseudoorbit and since f n (x) is an E-tracing point, it must be the unique one. But {y k } = {y k+n }, so it must be the case that x = f n (x), and hence x is periodic.

Stable and unstable sets and topological stability
We now introduce notions of stable and unstable sets.
In other words, U is wide if for any x not in the compact setṠ, the cross section U[x] is X.
Proof. Let x be any point in X and letḂ ⊂ Y be any neighborhood of f (x); to prove continuity we will find a neighborhoodȦ of x such Lemma 18. Let f ∈ E(X) and let A be a proper expansive neighbor- , which is compact, so cl L has a limit point, (p, q). Clearly p = q. On the other hand, choose a subsequence for all i and A is an expansive neighborhood, we must have that p = q, which is a contradiction.
For the converse, for each n with |n| ≤ N we have a neighborhood U n of ∆ X such that F (U n ) ⊂ A. Take U = |n|≤N U n . This completes our proof.
Lemma 19. If B is a proper expansive neighborhood for f and x is a periodic point, then W σ B (x) ⊂ W σ (x), where σ = s, u.
Proof. We will prove the case σ = s; the case σ = u is similar. For the sake of contradiction, suppose there is a periodic point x ∈ X and a proper expansive neighborhood B such that W s B (x) ⊂ W s (x). Say that x has period p and the periodic orbit is {x 0 , . . . , , there is a y ∈ W s B (x) and a neighborhood C of ∆ X such that F n (x, y) ∈ C for infinitely many n > 0. So, there exist infinitely many points of the form (x j , f n k (y)) for some 0 ≤ j < p−1. Because B[x j ] is compact, the sequence {(x j , f n k (y))} has a limit point (x j , y 0 ) ∈ B − int(C) (without loss of generality, assume that the sequence is convergent). Clearly, F n (x j , y 0 ) ∈ B for all n ≥ 0. For for all n < 0. By expansiveness, this implies that (x j , y 0 ) ∈ ∆ X . But (x j , y 0 ) ∈ int(C). So this is a contradiction. Proposition 20. Let f be topologically Anosov. Then we can find neighborhoods B and D of ∆ X and a continuous map t : D → X such that contains at most one point for any x, y ∈ X, Proof. Let A be an expansive neighborhood for f . Let B be a symmetric neighborhood of ∆ X such that B 3 ⊂ A, and let E = B ∩ F −1 (B), which is also a symmetric neighborhood of ∆ X . Since f has the shadowing property, there exists a neighborhood D of ∆ X such that every D-pseudo-orbit is E-traced by some orbit in X. We claim that E 2 is an expansive neighborhood for f , and hence by Lemma 15 the tracing point is unique. Let (x, y) ∈ E 2 . Then there is a z ∈ X such that (x, z), (z, y) ∈ E. So (f (x), f (z)), (f (z), f (y)) ∈ B and hence (f (x), f (y)) = F (x, y) ∈ B 2 ⊂ B 3 ⊂ A. This implies that F (E 2 ) is an expansive neighborhood for f , and hence so is E 2 . For each point (x, y) ∈ D, define a D-pseudo-orbit {x i } in X by Let t(x, y) denote the unique E-tracing point. This defines a map t : D → X (we postpone the proof of continuity to the end of the proof).
Let (x, y) ∈ D. Then F n (x, t(x, y)) ∈ E ⊂ B for all n ≥ 0, so t(x, y) ∈ W s B (x). Likewise, F n (y, t(x, y)) ∈ E ⊂ B for all n < 0 and since F (E) ⊂ B, F −1 (y, t(x, y)) ∈ E implies (y, t(x, y)) ∈ B. So t(x, y) ∈ W u B . In particular, t(x, y) ∈ W s B (x) ∩ W u B (y). Furthermore, the expansivity of f does not allow W s B (x) ∩ W u B (y) to have more than one point. This proves (1) and (2).
We prove (3); the proof of (4) is similar. To prove equality we prove containment in both directions. (⊃): Let y = t(x, y) and (x, y) ∈ D.
Then y E-traces x in forward time. So F n (x, y) ∈ E ⊂ B for all n ≥ 0.
. Then, F n (x, y) ∈ B for all n ≥ 0. Clearly F n (y, y) ∈ B for all n < 0, so y B-traces the pseudo-orbit {x i } (defined as above). By definition, the point t(x, y) E-traces this pseudo-orbit, and because E ⊂ B, it also Btraces it. So, for all n ∈ Z, (f n (t(x, y), x n )) ∈ B and (f n (y), x n ) ∈ B, which implies that F n (t(x, y), y) ∈ B 2 ⊂ A. Thus, by expansiveness, y = t(x, y).
We now show that the tracing map t is continuous. Let U be a wide neighborhood of ∆ X . By Lemma 17 we must find a neighborhood V of ∆ X such that T (V ) ⊂ U, where T = t × t. By Lemma 18, we can find N such that V N (A) ⊂ U. Define neighborhoods of ∆ X Let g : X 4 → X 4 be the homeomorphism given by g(x, y, x 1 , y 1 ) = (x, x 1 , y, y 1 ), Because (x, y), (x 1 , y 1 ) ∈ D, F n (t(x, y), x), F n (x 1 , t(x 1 , y 1 )) ∈ B for 0 ≤ n ≤ N, and because (x, x 1 ) ∈ W 1 , F n (x, x 1 ) ∈ B for 0 ≤ n ≤ N. So F n (t(x, y), t(x 1 , y 1 )) ∈ B 3 ⊂ A for 0 ≤ n ≤ N. Similarly, we can show that F n (t(x, y), t(x 1 , y 1 )) ∈ A for −N ≤ n ≤ 0, and hence T (V ) ⊂ V N (A) ⊂ U. Thus t is continuous.
Definition 21. A homeomorphism f : X → X is called topologically stable if for any neighborhood B of ∆ X there exists a neighborhood D of ∆ X such that for any homeomorphism g : X → X satisfying (f (x), g(x)) ∈ D, for all x ∈ X there exists a continuous self map h of X satisfying (h(x), x) ∈ B, for all x ∈ X and f • h = h • g.
We close this section by proving the following extension of a main result in [17].
Theorem 22. If X is a first countable locally compact paracompact Hausdorff space and f : X → X is topologically Anosov, then f is topologically stable.
Proof. Let B be a neighborhood of ∆ X . Let A be a proper expansive neighborhood for f and E be a symmetric neighborhood of ∆ X such that E 3 ⊂ A ∩ B. By Lemma 15 there is a neighborhood D of ∆ X such that every D-pseudo-orbit is uniquely E-traced by some point of X. Now take a homeomorphism g : X → X which is D-close to f . Let x ∈ X. Because (f (g n−1 (x)), g n (x)) ∈ D for all n ∈ Z, {g n (x) : n ∈ Z} is a D-pseudo-orbit for f . By the shadowing property, this defines a map h : X → X sending x to the unique point h(x) that E-traces the pseudo-orbit.
Remark 23. In fact, we can prove something slightly stronger than the conclusion of the previous theorem. If the neighborhood B is symmetric and B 2 is an expansive neighborhood for f , then h is unique. Suppose h ′ is another continuous map satisfying the same properties as h.
Remark 24. If the homeomorphism g in the above theorem is also expansive with B 2 an expansive neighborhood and B is symmetric, then h is injective. Indeed, suppose h(x) = h(y) for some x, y ∈ X. Then for any n ∈ Z, h(g n (x)) = f n (h(x)) = f n (h(y)) = h(g n (y)).

Chain recurrence and decomposition theorem
Chain recurrence is an important notion in dynamical systems defined on metric spaces. In this section we introduce a new, topological generalization of chain recurrence. We then prove our main theorem, Theorem 3.
Definition 25. Let x, y ∈ X. If there is an A-chain from x to y and one from y to x for every neighborhood A of ∆ X , then we write x ∼ y. Proof. It is clear from the definitions that Ω(f ) ⊂ CR(f ), so we must show the opposite inclusion. Let x ∈ CR(f ) andU be an open neighborhood of x. Let E be a neighborhood of ∆ X such that E[x] ⊂U. By the shadowing property there exists a neighborhood D of ∆ X such that every D-pseudo-orbit is E-traced by some point. Since x is chain recurrent, there exists a D-chain {x 0 = x, x 1 , . . . , x n = x}. We can extend this D-chain (in any way we want) to a full D-pseudo-orbit. This pseudo-orbit is E-traced by a point y ∈ X. This means that y, f n (y) ∈ E[x] ⊂U , and so x ∈ Ω(f ).
We have the following generalization of a result proved by Yang ([18, Lemma 1(2)]); Yang uses the metric definitions of expansivity, shadowing, and chain recurrence.
Proposition 28. If f is topologically Anosov, then Per(f ) is dense in CR(f ).
Proof. Let x ∈ CR(f ) andU be an open set containing x. Let E be a neighborhood of ∆ X such that E[x] ⊂U and E 2 is an expansive neighborhood for f . It suffices to show that there is a periodic point in E[x].
Proposition 29. Let f : X → X be a homeomorphism and R be a chain component of f . Then, Proof. First we will prove that the set is a closed set in X ×X. Let {(x i , y i )} ∞ i=0 ⊂ S be a sequence converging to (x, y) ∈ X × X. Let A be a neighborhood of ∆ X ; we must prove that there is an A-chain from x to y.
(1) The diagonal ∆ X is closed in X × X. Thus, so is CR(f ) = π 1 ∆ X ∩S , where π 1 : X × X → X is the projection onto the first coordinate.
(2) Given any x ∈ X,S[x] is closed, andS[x] is the chain component of f containing x.
(3) Let x ∈ R. We must show that f (x) ∈ R. Let A be a neighborhood of ∆ X . Clearly {x, f (x)} is an A-chain from x to f (x). Thus it remains to find an A-chain from f (x) to x. Choose neighborhoods B and C of ∆ X such that B 2 ⊂ A and C ⊂ B ∩ f −1 (B). Since x is chain recurrent, there is a C-chain {x = x 0 , x 1 , . . . , x = x n } from x to itself. Then {f (x), x 2 , x 3 , . . . , x = x n } is the desired A-chain.
Proposition 30. If f is topologically Anosov and R is a chain component, then R is both open and closed in CR(f ). ∩ CR(f ) is in R. Let E be a neighborhood of ∆ X ; we will find an E-chain from x to x via a point of R.
Let U be a neighborhood of ∆ X such that U 2 ⊂ E ∩ D. By Proposition 28, there is a periodic point p ∈ U[x] ⊂ D[x]. Then there exists q ∈ R such that (p, q), (q, p) ∈ D; by Proposition 28 we may assume q is periodic. By Proposition 20(2) and Lemma 19, there exist z 1 , z 2 ∈ X such that z 1 ∈ W u (p) ∩ W s (q) and is a U-chain, and hence is an E-chain.
Definition 31. A map f : X → X is topologically transitive if for any pair of non-empty open setsĠ,Ḣ ⊂ X, there exists n ∈ N such that f n (Ġ) ∩Ḣ = ∅.
We now show that a map that is expansive and has the shadowing property is topologically transitive on each chain component. This is a generalization of a result of Yang's ( [18,Lemma 3]) in which he uses the metric definitions of expansivity, shadowing, and chain recurrence.
Proposition 32. If f is topologically Anosov and R is a chain component, then f | R : R → R is topologically transitive.
Proof. LetĠ andḢ be nonempty open sets in R. Let x ∈Ġ and y ∈Ḣ. Let E be a neighborhood of ∆ X such that x ∈ E[x]∩R ⊂Ġ, y ∈ E[y]∩R ⊂Ḣ, E[x]∩CR(f ) ⊂ R, and E 2 is an expansive neighborhood for f . By Lemma 15, there exists a neighborhood D of ∆ X (we may also assume D ⊂ E) such that every periodic D-pseudo-orbit is Etraced by a periodic orbit. Since x, y ∈ R, there is a D-chain from x to itself through y, {x 0 = x, . . . , x n = y, . . . , x m = x} in X which we can extend to a periodic D-pseudo-orbit, {. . . , x m−1 , x m = x 0 , . . . , x m = x 0 , x 1 , . . .}. By Lemma 15 there is a periodic point p ∈ X that Etraces this pseudo-orbit. In fact, Finally, we have all the ingredients to prove our main theorem.

Relationships to metric definitions
In this section with discuss the relationships between our topological definitions and the usual metric definitions for expansiveness, shadowing, and chain recurrence. First, we give another generalization of chain recurrence, due to Hurley.
Definition 33 ([8]). Let (X, d) be a metric space and f : X → X be a homeomorphism. Let P denote the set of continuous functions from X to (0, ∞). For δ(x) ∈ P, a δ(x)-chain is a sequence {x 0 , x 1 , . . . , x n } (n ≥ 1) such that d(f (x i−1 ), x i ) < δ(f (x i−1 )) for i = 1, . . . , n. A point x is strongly chain-recurrent if for any δ ∈ P, there is a δ(x)-chain from x to itself.
In the case that we have a metric space, the various properties are related in the following way.
Proposition 34. Let (X, d) be a metric space and f : X → X be a homeomorphism.
(2) Metric shadowing does not imply topological shadowing, nor does topological shadowing imply metric shadowing. (3) Strong chain recurrence and topological chain recurrence are equivalent. They imply metric chain recurrence, but not vice versa.
If X is compact, then metric and topological expansivity are equivalent, metric and topological shadowing are equivalent, and metric, topological, and strong chain recurrence are equivalent.
Proof. (1) If e is an expansive constant, then U e = d −1 [0, e) is an expansive neighborhood; thus metric expansivity implies topological expansivity. It is not difficult to see that the homeomorphism in Example 6 is topologically expansive. Thus, the opposite implication does not hold.
(2) The first half follows from Example 7. The given homeomorphism has the shadowing property with respect to one metric, but it does not have the topological shadowing property (this is easiest to see with the second metric).
Let f be the identity map on the space X = {x n } ∞ n=1 , where x n = n i=1 (1/i), given the metric inherited from R. Since X has the discrete topology, we can pick a neighborhood D of ∆ X such that D[x] = {x} for all x. Then a D-pseudo-orbit is an actual f -orbit (that is, a fixed point). So the topological shadowing property is trivially satisfied. On the other hand, given δ > 0 and N > 1/δ, {. . . , x N , x N , x N , x N +1 , x N +2 , . . .} is a δ-pseudo-orbit, but clearly such a pseudo-orbit is not traced by any orbit. So f does not have the metric shadowing property.
(3) Topological chain recurrence clearly implies strong chain recurrence. To prove the opposite implication, we must show that for any open neighborhood U of ∆ X , there exists a δ ∈ P such that B δ ⊂ U, where B δ = {(x, y) : d(x, y) < δ(x)}. We may assume that U[x] X for all x. Define a function h : X → (0, ∞) by h(x) = d(x, X − U[x]). Since h is lower semicontinuous, the function δ(x) = inf{f (y) + d(x, y) : y ∈ X} is continuous (see [19, Theorem 67.2]), and 0 < δ(x) < h(x) for all x. Thus B δ ⊂ B h ⊂ U.
It is clear that topological and strong chain recurrence imply metric chain recurrence. Hurley gives an example ([8, Example 1]) showing that the opposite implication does not hold.
Finally, the equivalences in the compact case are clear, as in Remark 11.
Hurley has shown ([8]) that strong (or topological) chain recurrence is the right notion to extend Conley's fundamental theorem of dynamical systems to noncompact metric spaces. The second example from (2) is expansive but not metric expansive, and it has the shadowing property but not the metric shadowing property; thus Theorem 3 applies, but a version using the usual metric definitions would not. These examples, together with Proposition 13, suggest that the topological definitions are a useful way to study the dynamical structure of a homeomorphism on a noncompact space, even if the space is compact and the usual metric definitions could be applied.