TOPOLOGICAL PROPERTIES OF INVARIANT SETS FOR ANOSOV MAPS WITH HOLES

TODD FISHER AND SKYLER SIMMONS

Abstract. We study topological properties of invariant sets of Anosov diffeo- morphisms with holes. Results related to cardinality, local maximality, , and dimension are presented. The main result states that the Hausdorff dimen- sion of the invariant set can be computed by the entropy of the invariant set together with the hyperbolicity constants.

1. introduction Physical systems are never truly isolated from the environment, so there is something that is escaping or leaking from the system. An open dynamical sys- tem encapsulates this phenomenon by allowing an iterate of a point to leave the state space. These systems lend themselves very naturally to certain physical phe- nomena; prominent examples of application in physics and chemisty include the escaping of a gas from a container (as in [1], [5], and [6]) or modeling sub-atomic quantum phenomena (as in [2] and [11]). Applications from other fields such as ecology have also been investigated [8]. A class of open systems that have been studied extensively are billiards. In the case of an one considers the set of points whose is bounded. In [13] Lopes and Markarian show that for a certain outer billiard the set of all points whose forward and reverse orbits hit the scatterers infinitely many times for forward and reverse time form a Cantor set, and the iteration map restricted to this invariant set is similar to Smale’s . This last example hints at the following very natural question: Question 1.1. For an open what can be said about the set of points whose orbits never leave the system? The objective of the present work is to address the above question for certain open systems. Specifically, we let the state space be a compact, connected, bound- aryless, smooth and the evolution be given by a diffeomorphism from the manifold to itself. We then fix an open set in the manifold and look at the set of points whose orbits never intersect the open set. These points will naturally form a compact f-invariant set in the manifold, and we examine properties for this

2000 Subject Classification. 37D05, 37D20, 37D35, 37B40. Key words and phrases. Open Systems, Hausdorff Dimension, Box Dimension, Anosov Map . 1 2 TODD FISHER AND SKYLER SIMMONS invariant set related to cardinality, local maximality, connectedness, entropy and Hausdorff dimension. Additionally, we will investigate what happens as the size and/or shape of the hole changes. Throughout the paper we assume the diffeomorphism is Anosov. A diffeomor- phism is Anosov if the entire manifold is a for a diffeomorphism f. If A ∈ GL(n, Z) where |det(A)| = 1 and no eigenvalues of A lie on the unit circle, then the hyperbolic toral automorphism fA induced by A is Anosov. A number of people have investigated this same class of open systems, looking at measure theoretic properties. A few of these results can be found in [3, 4]. One property that can be used in describing invariant sets is dimension. A direct application of dimension studies to open systems is given in a two-part paper by Horita and Viana (see [9] and [10]). The main result of the present work is the following:

Theorem 1.2. Let fA be a two-dimensional hyperbolic toral automorphism, and let λ be the absolute value of the eigenvalue of A greater than 1. Then if ΛU is an fA-invariant set associated with a hole U where U is simply connected and the boundary is a piecewise smooth , then the box dimension and Hausdorff dimension of ΛU is given by 2h (f | ) top A ΛU . log λ Using results of Chernov and Markarian [3, 4] it is known that if diam(U) → 0, then the limit of the Hausdorff dimension of ΛU approaches 2. The remainder of the paper proceeds as follows: In Section 2 we provide back- ground material on hyperbolicity, symbolic systems, entropy, and dimension. In Section 3 we prove results relating to topological properties of invariants sets for open systems. In Section 4 we prove Theorem 1.2.

2. background For a dynamical system (X, f) a set Λ ⊂ X is invariant if f(Λ) = Λ. For a point x ∈ X, the forward orbit of x is O+(x) := {f n(x): n ∈ N ∪ {0}}, and the orbit of x is O(x) := {f n(x): n ∈ Z}. A map f : X → X is said to be transitive if the forward orbit of some point x ∈ X is dense, and the map is topologically if given two open sets U, V ⊂ X, n there is some positive integer n0 such that for all n ≥ n0 we have f (U) ∩ V 6= ∅. Two maps f : X → X and g : Y → Y are topologically semi-conjugate if there is a continuous surjection k : X → Y such that k ◦ f = g ◦ k. The map k is said to be a topological semi-conjugacy. If k is a homeomorphism we call it a and say the dynamical systems (X, f) and (Y, g) are topologically conjugate. ANOSOV MAPS WITH HOLES 3

2.1. Hyperbolic Dynamics. Let M be a manifold, and f : M → M be a diffeomorphism. An invariant set Λ is a hyperbolic set if there is a splitting of u s TΛM = E ⊕ E and constants λ > 1 and C > 0 satisfying: n s n s s s •| Df (p)v | ≤ Cλ |v | for all v ∈ Ep, and −n u n u u u •| Df (p)v | ≤ Cλ |v | for all v ∈ Ep . If Λ = M, then f is an Anosov diffeomorphism. Let f : M → M be a diffeomorphism and Λ be a hyperbolic set for f. For  > 0 sufficiently small and x ∈ Λ the local stable and unstable are respectively: s n n W (x, f) = {y ∈ M | for all n ∈ N, d(f (x), f (y)) ≤ }, and u −n −n W (x, f) = {y ∈ M | for all n ∈ N, d(f (x), f (y)) ≤ }. The stable and unstable manifolds are respectively: s S −n s n W (x, f) = n≥0 f (W (f (x), f)) , and u S n u −n W (x, f) = n≥0 f (W (f (x), f)) . For a Cr diffeomorphism f the stable and unstable manifolds of a hyperbolic set are Cr injectively immersed submanifolds. For p ∈ Λ a point q ∈ M is a homoclinic point for p if q ∈ W s(p) ∩ W u(p), and is a transverse homoclinic point for p if s u q ∈ W (p) t W (p). A useful property for hyperbolic invariant sets is local maximality. A hyperbolic invariant set Λ is locally maximal if there exists an open set U ⊃ Λ such that \ Λ = f n(U). n∈Z An equivalent condition for local maximality is that of a local product structure. A hyperbolic set Λ for a diffeomorphism f : M → M has a local product structure s u if there exists δ > 0 and  > 0 such that W (p) ∩ W (q) consists of one point in Λ for all p, q ∈ Λ where d(p, q) < δ, see for instance [12, p. 272].

2.2. Markov Partitions. For Λ a hyperbolic set we know that if  is sufficiently s u small and x, y ∈ Λ, then W (x) ∩ W (y) consists of at most one point. For such an  > 0 define s u D = {(x, y) ∈ Λ × Λ | W (x) ∩ W (y) ∈ Λ} s u and [·, ·]: D → Λ so that [x, y] = W (x) ∩ W (y).

Definition 2.1. A rectangle is a closed set R ⊂ Λ such that R × R ⊂ D. For R a rectangle and x ∈ R denote s s u u W (x, R) = R ∩ W (x) and W (x, R) = R ∩ W (x). A rectangle R is proper if R = intR. 4 TODD FISHER AND SKYLER SIMMONS

Definition 2.2. Let Λ be a hyperbolic set for a diffeomorphism f : M → M and fix  sufficiently small. A finite cover R of Λ by proper rectangles with diam(R) <  for any R ∈ R is a Markov Partition if Ri,Rj ∈ R, x ∈ intRi, and f(x) ∈ intRj, then s −1 u • f(W (x, Ri)) ⊂ Rj and f (W (f(x),Rj)) ⊂ Ri, and • int(Ri) ∩ int(Rj) = ∅ if i 6= j. Associated with each of these partitions is a subshift of finite type and a semi- conjugacy to Λ. For the associated adjacency matrix A we denote the associated subshift of finite type by (ΣA, σA).

2.3. Entropy. The topological entropy of a system reflects the of the system. To be more precise. given a metric space X with metric d, continuous map f : X → X, and n ∈ N, define a distance function j j dn(x, y) = max (d(f (x), f (y)). 0≤j

A set S ⊂ X is said to be (n, )-separated for f if dn(x, y) >  for any distinct x, y ∈ S. Define

rsep(n, ) = max{#(S): S ⊂ X is an (n, )-separated set}, Then the topological entropy for f is   log(rsep(n, )) htop(f) = lim lim sup . →0 n→∞ n Alternatively, a set S ⊂ X is said to be (n, )-spanning for f if for any point x ∈ X there is a point y ∈ S such that dn(x, y) ≤ . Defining rspan(n, ) to be the smallest number of elements in an (n, )-spanning set, we can also define the entropy of the system as   log(rspan(n, )) htop(f) = lim lim sup . →0 n→∞ n In [14, p. 380] it is demonstrated that one can use spanning sets, separated sets, and open covers to define the entropy, and that all are equivalent. Another important result on topological entropy is that the entropy of a subshift of finite type with adjacency matrix A is the largest modulus of an eigenvalue of A. Topological entropy is one quantity that is preserved under conjugacy as well as under uniform finite-to-one semi-conjugacy. For an f-invariant Borel probability measure µ, the measure-theoretic entropy of f is denoted hµ(f). Roughly speaking, the measure-theoretic entropy measures the exponential growth rate of orbits which are “relevant” to µ. (A precise definition of measure-theoretic entropy can be found in [12, p. 169]) ANOSOV MAPS WITH HOLES 5

2.4. Dimension. Multiple definitions exist for computing the dimension of a set. We focus here on box dimension and Hausdorff dimension. For a compact set S ⊂ Rn, let N(, S) be the minimum number of -cubes covering S. Then the lower box dimension (or inner box dimension) of S is defined as log(N(, S)) dimb(S) = lim inf →0 − log() and the upper box dimension (or outer box dimension) of S is defined as log(N(, S)) dimB(S) = lim sup . →0 − log() When the two coincide, the limit is the box dimension of the set S. Alternatively, we can consider the Hausdorff dimension of a compact set S. d First, we define the d-dimensional Hausdorff Content of S, denoted CH (S). This is given by   d X d [ CH (S) = inf ri | S ⊂ Bri (xi) i i where the infimum is taken over all collections of balls covering S. The Hausdorff Dimension of a set S, denoted dimH (S), is the infimum of all non-negative numbers d such that the d-dimensional Hausdorff content of S is zero. If S ⊂ Rn is compact, we have

0 ≤ dimH (S) ≤ dimb(S) ≤ dimB(S) ≤ n.

3. Topological properties of invariant sets This section describes topological properties of invariant sets associated with open systems and describes the topological dynamics for these systems. Let f : M → M be Anosov and U ⊂ M be open. We denote the invariant set associated with this open system by n ΛU = {x ∈ M : f (x) ∈/ U, ∀n ∈ Z}. Our first result seems to be well known, but we could not find a proof of it in the literature. We provide a short proof since the result will be useful later in the paper. Claim 3.1. For any non-empty open set U a subset of the 2-torus, T2, and f : 2 2 T → T Anosov, the associated invariant set ΛU is totally disconnected. Proof. Let U be an arbitrary open set in M. By density of the periodic points for a transitive Anosov diffeomorphism we know that U contains a z and that the stable and unstable manifolds of z are dense in T2. Let x and y be distinct points of ΛU . Let Px and Py be two parallelograms whose u s sides are segments of W (z) and W (z) such that x ∈ Px, y ∈ Py, and Px ∩Py = ∅. (The last condition is possible by the density of W u(z) and W u(z).) Hence there 6 TODD FISHER AND SKYLER SIMMONS are two disjoint open sets in ΛU that contain one of x or y but not both, and ΛU is totally disconnected. 

3.1. Cardinality of Invariant Sets. We now present two results that give a relationship between the size of the set U and the cardinality of the set ΛU . Proposition 3.2. Let f : M → M be a transitive Anosov diffeomorphism. Then there is a positive number δ∞ such that if U is an open ball in a manifold M with ∞ radius smaller than δ , then the set ΛU contains infinitely many points.

Proof. Let y1, y2 be periodic points such that O(y1) ∩ O(y2) = ∅. Fix x1 a homoclinic point for y1 and x2 a homoclinic point for y2. Now, for every point x ∈ M, let d1 be the distance between x and O(x1) and let d2 be the distance between x and O(x2). Let rx be the greater of d1 and d2. (Note that by definition, O(x1) ∩ O(x2) = ∅, and both sets are closed, so rx > 0.) Define Vx to be an open ball about the point x of radius rx. Then the collection of all sets {Vx} is an open cover for M. Hence there is some finite subcover of these sets. Let δ∞ be the Lebesgue number for that finite subcover. Notice now that if U ⊂ M is an open ball of radius smaller than δ∞, then U ⊂ Vx for some x ∈ M. Then either O(x1) or O(x2) does not intersect U by construction of each set Vx. So ΛU contains a non-periodic point, and so ΛU has infinite cardinality.  Proposition 3.3. Let f : M → M be a transitive Anosov diffeomorphism. Then there is a positive number δfinite such that if U is an open ball with radius greater finite than δ , then the set ΛU is finite. The proof of this depends on two preliminary results:

Claim 3.4. For a closed (compact) set K, define dK (x) = d(x, K), where d(x, K) represents the distance function between x and K. Then |dK (x)−dK (y)| ≤ d(x, y). Proof. By the triangle inequality, we have d(x, K) ≤ d(x, y) + d(y, K) and d(y, K) ≤ d(x, y) + d(x, K). Hence, −d(x, y) ≤ d(y, K) − d(x, K) ≤ d(x, y) or

|d(x, K) − d(y, K)| = |dK (x) − dK (y)| ≤ d(x, y) as claimed.  Claim 3.5. Let J be an arbitrary collection of closed subsets of M. Define g(x) = supK∈J dK (x). Then g(x) is continuous. ANOSOV MAPS WITH HOLES 7

Figure 1. A of the torus

Proof. Let  > 0 be given, and let x be an arbitrary point of M. Then for all y with d(x, y) < /2, we know that |dK (x) − dK (y)| < d(x, y) for all K ∈ J by Claim 3.4. Then |g(x) − g(y)| ≤ /2 < . 

Proof of Proposition 3.3. Let S be a finite set of periodic points of M together with every point in their respective orbits. For every point x 6∈ S, define Kx = O(x), the closure of the orbit of x. Let J be the set of all Kx. Define the function g(x) as in Claim 3.5. Let δfinite be the maximum of g(x) on M. Then for any point x ∈ M, if U is an open ball of radius greater than δfinite, then U contains at least one point of the orbit of each point not in S. Hence, ΛU is a subset of S, and must therefore be finite.  For both Proposotions 3.2 and 3.3, no Riemannian metric was specified for the distance functions giving the size of the open sets and the point-set distances. Both results proceed without difficulty if any Riemannian metric is used. Furthermore, a specific computation of δfinite and δ∞ can be somewhat difficult. However, in the case of hyperbolic toral automorphisms we can establish bounds for these values by considering the cardinality of remaining points when rectangles are removed from a Markov partition. Remark 3.6. There are examples on the 2-torus for a hyperbolic toral automor- phism where the set ΛU is countable. To see this, consider the partition of T2 as given in Figure 1. This corresponds to a hyperbolic toral automorphism fA, where 1 1 A = . 1 0

If we let R1 be the larger unshaded rectangle, and R2 the shaded rectangle, then the transition matrix T for fA is given by T = A. Now, consider the effect of 8 TODD FISHER AND SKYLER SIMMONS removing the interior of R1. Since transitions from the interior of R2 to itself are impossible, the only points which can be in ΛR2 are the homographic points corresponding to the corners of R2. Hence, ΛR2 is a countable set. 3.2. Local maximality and transitivity. We now investigate the properties of local maximality and transitivity for ΛU . This investigation is relevant for our later analysis since many results relating to hyperbolic sets assume that the set is locally maximal and/or transitive. One may hope that for sufficiently small holes the set ΛU satisfies at least one of these properties; however, that is not the case. Let f : M → M be an Anosov diffeomorphism and suppose that R is a Markov partition of M for f such that there is a rectangle R with a fixed point, p, on the boundary of the rectangle. Examples where this is the case can be constructed in any dimension using methods described by Sinai [15]. Now let U be the interior of s u the rectangle R. Then the set ΛU will contain sequences {qk} and {qk } contained in the stable and unstable manifold of p, respectively, that converge to p and are contained in ΛU . However, for k sufficiently large and  sufficiently small we u s s u know that W (qk) ∩ W (qk ) is not contained in ΛU . So ΛU does not have a local product structure and therefore is not locally maximal. The previous example of a countable invariant set falls into this category. In regards to transitivity the authors have found examples where the holes are given by the interior of an arbitrarily small rectangle from a Markov partition and the resulting invariant set is not transitive.

4. The dimension of ΛU In this section we prove the main theorem and investigate the Hausdorff di- mension and box dimension of ΛU . Results in this section will be limited to the two-dimensional case and we will require the set U to be simply connected.

Proof of Theorem 1.2. Let fA be a two-dimensional hyperbolic toral automorphism, and let λ be the absolute value of the eigenvalue of A greater than 1. Let the metric d be given by the ∞-metric in terms of the basis given by the two eigenvectors of A. In this metric, a an open ball is a square with sides parallel to the eigenvectors of A. Let U be a simply connected open set in the torus whose boundary is piecewise smooth. Define ΛU to be the set of points in the torus that never enter U under iteration. Let  > 0 such that 1/λ +  < 1. If B(x) represents the open ball about x of radius  in the metric d, then for n ∈ N define B(x, n) to be n \ −i i B(x, n) = fA B(fA(x)). i=−n

Note that this is equal to Bλ−n (x) by our choice of metric. Define N (n, ) to be the minimum number of sets B(x, n) required to cover ΛU . Before proceeding ANOSOV MAPS WITH HOLES 9 with the proof we need the following estimate.

Lemma 4.1. Let rspan(n, ) denote the size of a minimal (n, )-spanning set for ΛU . Then N (n, ) ≤ rspan(2n + 1, ).

Proof. Let Y be a minimal (2n + 1, )-spanning set for ΛU . Then for all x ∈ ΛU there is a point y of Y such that d(f i(x), f i(y)) <  for all 0 ≤ i ≤ 2n + 1. But then it certainly holds that d(f i(x), f i(y)) <  for all 0 ≤ i ≤ 2n, and so Y is a (2n, )-spanning set for ΛU as well.

n Now, consider the set {Bn(f (y), ): y ∈ Y }. We claim that this forms a cover for the set ΛU . To see this note that n i n i Bn(f (y), ) = {x ∈ ΛU : d(f (f (y)), f (x)) <  : −n ≤ i ≤ n} i i−n = {x ∈ ΛU : d(f (y)), f (x)) <  : 0 ≤ i ≤ 2n}, where the second equality follows from f-invariance. Since Y is a (2n, )-spanning n set and ΛU is invariant, every point x ∈ λU is included in one of the sets Bn(f (y), ). n Hence, the collection {Bn(f (y), ): y ∈ Y } is a cover for ΛU . As this cover may not be minimal, we get N (n, ) ≤ rspan(2n + 1, ).  We now return to the proof of Theorem 1.2. By definition, we have   log(rspan(2n + 1, )) htop(fA|ΛU ) = lim lim sup →0 n→∞ 2n + 1 log(N (n, )) ≥ lim lim sup →0 n→∞ 2n + 1 with the inequality in Lemma 4.1. Multiplying both sides by 2 yields log(N (n, )) log(N (n, )) 2htop(fA|Λ ) ≥ lim lim sup = lim lim sup . U →0 1 →0 n→∞ n + 2 n→∞ n

Let N(, ΛU ) be the minimum number of balls of radius  that are needed to cover ΛU , as in the definitions for upper and lower box dimensions. Recall that the definition of upper box dimension is

log(N(, ΛU )) dimB(ΛU ) = lim sup . →0 − log() Since N (n, ) measures the number of balls having radius λ−n required to cover ΛU , we can also write the box dimension in terms of these quantities as follows: log(N (n, )) log(N (n, )) log(N (n,)) dim (Λ ) = = = n . B U − log(λ−n) − log() + n log(λ) − log() n + log(λ) 10 TODD FISHER AND SKYLER SIMMONS

We then know that log(N (n,)) n dimB(ΛU ) = lim lim sup − log() →0 n→∞ n + log(λ) lim lim sup log(N (n,)) = →0 n→∞ n log(λ) 2h (f | ) ≤ top A K , log(λ) which establishes an upper bound for the box dimension. We now want to find the Hausdorff dimension of the set ΛU . The main tool will be the following theorem due to Young. (We refer to [16] for relevant defintions.) Theorem 4.2. [16] Let f : M → M be a C2 diffeomorphism of a compact surface and µ be an ergodic Borel probability measure with Lyapunov exponents λ1 ≤ λ2. Then  1 1  dimH (µ) = hµ(f) − λ1 λ2 whenever the right side of the equation is not of the form 0/0. The following result is an immediate corollary to the theorem above. Corollary 4.3. Let Λ be an invariant set for a hyperbolic toral automorphism 2 2 fA : T → T and µ be a measure of maximal entropy for Λ. If λ is the modulus of the expanding eigenvalue for the matrix A, then  1 1  2h (f | ) dim (µ) = h (f | ) − = top A Λ . H top A Λ log(λ) log(1/λ) log(λ) Note that we are not assuming that there is a unique measure of maximal entropy. A reason that the result above is helpful is that it does not rely on properties such as transitivity or local maximality. Theorem 1.2 now follows since 2h (f | ) 2h (f | ) top A ΛU = dim (µ) ≤ dim (Λ ) ≤ dim = top A ΛU . log(λ) H H U B log(λ)  References 1. H. Alt, H.-D. Gr¨af,H. L. Harney, R. Hofferbert, H. Rehfeld, A. Richter, and P. Schardt, Decay of classical chaotic systems: the case of the Bunimovich stadium, Phys. Rev. E (3) 53 (1996), no. 3, 2217–2222. MR 1388211 (97c:58094) 2. G. Casati, Quantum relaxation and Poincar´erecurrences, Phys. A 288 (2000), no. 1-4, 49–60. MR 1808174 (2001k:81074) 3. N. Chernov and R. Markarian, Ergodic properties of Anosov maps with rectangular holes, Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2, 271–314. MR 1479505 (98k:58140) ANOSOV MAPS WITH HOLES 11

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T. Fisher, Department of Mathematics, Brigham Young University, Provo, UT 84602 E-mail address: [email protected]

S. Simmons, Department of Mathematics, Brigham Young University, Provo, UT 84602 E-mail address: [email protected]