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, entropy, 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 outer billiard one considers the set of points whose orbit 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 horseshoe map. This last example hints at the following very natural question: Question 1.1. For an open dynamical system 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 manifold 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 Mathematics 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 hyperbolic set for a diffeomorphism f. If A 2 GL(n; Z) where jdet(A)j = 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 curve, then the box dimension and Hausdorff dimension of ΛU is given by 2h (f j ) 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 2 X, the forward orbit of x is O+(x) := ff n(x): n 2 N [ f0gg; and the orbit of x is O(x) := ff n(x): n 2 Zg. A map f : X ! X is said to be transitive if the forward orbit of some point x 2 X is dense, and the map is topologically mixing 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 topological conjugacy 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 •j Df (p)v j ≤ Cλ jv j for all v 2 Ep, and −n u n u u u •j Df (p)v j ≤ Cλ jv j for all v 2 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 2 Λ the local stable and unstable manifolds are respectively: s n n W (x; f) = fy 2 M j for all n 2 N; d(f (x); f (y)) ≤ g; and u −n −n W (x; f) = fy 2 M j for all n 2 N; d(f (x); f (y)) ≤ g: 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 2 Λ a point q 2 M is a homoclinic point for p if q 2 W s(p) \ W u(p), and is a transverse homoclinic point for p if s u q 2 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): n2Z 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 2 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 2 Λ, then W (x) \ W (y) consists of at most one point. For such an > 0 define s u D = f(x; y) 2 Λ × Λ j W (x) \ W (y) 2 Λg 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 2 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 2 R is a Markov Partition if Ri;Rj 2 R, x 2 intRi, and f(x) 2 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 complexity of the system. To be more precise. given a metric space X with metric d, continuous map f : X ! X, and n 2 N, define a distance function j j dn(x; y) = max (d(f (x); f (y)): 0≤j<n A set S ⊂ X is said to be (n; )-separated for f if dn(x; y) > for any distinct x; y 2 S. Define rsep(n; ) = maxf#(S): S ⊂ X is an (n; )-separated setg; Then the topological entropy for f is log(rsep(n; )) htop(f) = lim lim sup : !0 n!1 n Alternatively, a set S ⊂ X is said to be (n; )-spanning for f if for any point x 2 X there is a point y 2 S such that dn(x; y) ≤ .