Open maps: small and large holes with unusual properties

Let $X$ be a two-sided subshift on a finite alphabet endowed with a mixing probability measure which is positive on all cylinders in $X$. We show that there exist arbitrarily small finite overlapping union of shifted cylinders which intersect every orbit under the shift map. We also show that for any proper subshift $Y$ of $X$ there exists a finite overlapping unions of shifted cylinders such that its survivor set contains $Y$ (in particular, it can have entropy arbitrarily close to the entropy of $X$). Both results may be seen as somewhat counter-intuitive. Finally, we apply these results to a certain class of hyperbolic algebraic automorphisms of a torus.


Introduction
This paper is concerned with an area of dynamics which is usually referred to as "maps with holes", "open maps" or "open dynamical systems". Let X be a compact (or precompact) metric space and f : X → X be a map with positive topological entropy. Let H ⊂ X be an open set which we regard as a hole.
In this area of dynamics, we consider the set of points that do not fall into a "hole" under iterations of a map. This is a well studied area with numerous papers. The first paper in this area was [18], where they considered a game of billiards with a hole somewhere in the table. The rate at which a random billiard ball "escaped" was considered, and depended upon both the size and the location of the hole. It is clear that if a hole is enlarged, then the rate of escape will not decrease, and often increase. In [1,5], it was shown that the location could sometimes play a more significant role in the escape rate than the size of the hole. An important factor in the escape rate is the set of periodic points of the map that fall into the hole [3]. For a good history of these problems, see [9,10,11].
The problem that we study, while related, is not exactly the same. We are concerned with the set of points that will always avoid a hole, that is, that will never escape. The precise connection between the theory of escape rates and our results is explained in detail in Remark 4.16 at the end of the paper.
Formally, we denote by J (H) the set of all points in X whose f -orbit does not intersect H and call it the survivor set. Clearly, a survivor set is f -invariant, and in a number of recent papers certain dynamical properties of the map f | J (H) have been studied -see, e.g., [2] and references therein.
It seems that a more immediate issue here is the "size" of a survivor set. At first sight, it would seem plausible that if H is "large", then J (H) is countable -or even empty. On the other hand, if it is "small", then one might expect the Hausdorff dimension of J (H) to be positive.
The starting point for this line of research has been the case when X = [0, 1] and T (x) = 2x mod 1, the doubling map with T (1) = 1. Assume our hole to be connected, so we have H = (a, b) (0, 1). We denote J (H) by J (a, b).
and this bound is sharp.
. 1 A similar claim holds for the β-transformation x → βx mod 1 for β ∈ (1, 2) -see [7]. Thus, in the one-dimensional setting one's naive expectations prove to be spot on.
The situation is however very different for the baker's map. Namely, put X = [0, 1] 2 ; the baker's map B : X → X is the natural extension of the doubling map, conjugate to the shift map on the set of bi-infinite sequences. We have The purpose of this note is to extend these two results from the full shift on two symbols to more general subshifts (Theorems 2.4 and 3.1) and apply these to the generalized Pisot toral automorphisms (Section 4). 1 If we regard the doubling map as the map z → z 2 on the unit circle S 1 , then 1 2 needs to be replaced with 2 3 , in which case J (a, b) ⊂ {1}.

Symbolic model: small holes
Let A be a finite alphabet. Let σ : A N → A N denote the left shift, i.e., We define distance on A N with A = {0, 1, . . . , k − 1} in the usual way as This topology is equivalent to the product topology. We say that X ⊂ A N is a subshift if X is closed under this topology, and σ(X) = X. Suppose µ is a probability σ-invariant measure on X which we assume to be positive on all cylinders. We denote the set of admissible words of length n for X by L n (X) and put L(X) = n≥0 L n (X), the language of X.
The topological entropy of a subshift X is defined by the formula Recall that a subshift (X, σ) is called irreducible if, for every ordered pair of words u and v ∈ L(X), there is a w ∈ L(X) with uwv ∈ L(X). We say a subshift is sofic if there exists a regular language (i.e., a language accepted by a finite automaton) such that X is the set of all infinite sequences that do not contain a subword from this regular language. For more detailed see [15,Chapter 3]. An irreducible sofic subshift is known to have a unique measure of maximal entropy µ (see [26]), which has the following property (see [13,Lemma 4.8]): there exists θ ∈ (0, 1) such that . . w 0 .w 1 . . . w n ] ≍ θ n , ∀w −n+1 . . . w n ∈ L 2n (X).
Recall that µ is called non-atomic if µ({x}) = 0 for any x ∈ X. We say that a measure µ is mixing if for any pair of cylinders as n → +∞. Similarly to the case of the baker's map, we say that a Borel A ⊂ X is a complete trap if it intersects all orbits in X, i.e., for any x ∈ X there exists n ∈ Z such that σ n x ∈ A.
Remark 2.1. Notice that a mixing measure positive on all cylinders is always non-atomic. Indeed, assume that µ(A) > 0, where A = {x}. Then by mixing, µ(σ −n A ∩ A) → µ(A) 2 , which is impossible unless x is a fixed point for σ. If x is such, then it must have x i ≡ j for all i ∈ Z and we have µ(A) = 1, which is impossible, since µ cannot be supported by a single orbit, being positive on all cylinders.
Remark 2.2. A similar notion exists in Combinatorics on Words, namely, unavoidable sets. A subset S of A * is called unavoidable if any word in A * contains an element of S as a factor. These have been introduced by M.-P. Schützenberger in [21], and various results, mostly regarding their size, have been proved since -see, e.g., [6] and references therein.
In the case of [0, 1] and [0, 1] 2 in Section 1, we found connected components with unusual properties. If X is a subshift, then X is totally disconnected, so X has no non-trivial connected components. Instead we consider what we believe is a natural analog of connected, in the case of subshifts. That is, being a finite union of overlapping shifted cylinders. More formally We say that C is overlapping if for all i and j there exists The main result of this section is that for each two-sided subshift with mild assumptions there exist arbitrarily small overlapping complete traps. More precisely: Theorem 2.4. Let X be a two-sided subshift on a finite alphabet A endowed with a shiftinvariant probability measure µ on X which we assume to be positive on all cylinders and mixing (hence non-atomic) in X.
Then for any ε > 0 there exists a finite union of shifted cylinders C : This will be proven by a series of lemmas.
Proof. It is worth observing that the σ −r i [w i ] may overlap, and the w i may have different lengths. For N ≥ |w i | + r i , we see that we can find a prefix u of length r i and suffix v of length N − r i − |w i | such that uw i v ∈ L(X) is a word of length N. We let S be the set of all such uw i v for all pairs w i and r i . This gives us a set C = w ′ i ∈S [w ′ i ] of disjoint union of cylinders of equal length. The fact that we can choose N sufficiently large so that µ([w]) < ε follows from Lemma 2.5.
Lemma 2.7. Let X be a two-sided subshift on a finite alphabet A endowed with a shiftinvariant probability measure µ on X which we assume to be positive on all cylinders and mixing (hence non-atomic) in X. Let C = [w i ] be a finite union of disjoint cylinders and a complete trap. Then there exists r i ≥ 0 such that such that Proof. As µ is positive on all cylinders, we see that µ([w i ]) > 0 for all choices i. For all w i and w j there will exist a M i,j such that for all Proof. If x 1 ≥ 1 4 , then we take k = 1. Otherwise let k be such that k−1 Lemma 2.9. Let X be a two-sided subshift on a finite alphabet A endowed with a shiftinvariant probability measure µ on X which we assume to be positive on all cylinders and mixing (hence non-atomic) in X.
Then for any ε > 0 there exists a C := [w i ], a finite union of disjoint cylinders such that (2) C is a complete trap.
Proof. Put C 0 = a∈A [a]. Clearly C 0 = X, and hence is a complete trap.
We proceed by induction. Assume we have a finite collection of cylinders C n = σ −r i [w i ] which is a complete trap. Using Lemma 2.6 we write C n = [w ′ i ], a disjoint union of cylinders where µ([w ′ i ]) < µ(C n )/2 and all w ′ i are the same length, say N. By Lemma 2.8 and the fact that all cylinders have µ([w ′ ]) < µ(C n )/2, we can partition the set of cylinders into two sets C ′ n and C ′′ n where Put t n = µ(C n ). Then we have Clearly, t n is decreasing and positive; let L = lim n→∞ t n . Then L ≤ L − L 2 /32, whence L = 0. Choose n such that µ(C n ) < ε and use Lemma 2.6 to write C := C n . This union in the desired form.
Remark 2.10. It follows from the proof of Lemma 2.9 that µ being mixing can be replaced with the following, weaker, condition: there exists a δ > 0 such that for all cylinders -provided we assume that µ({x}) = 0 for any fixed point x ∈ X.
Remark 2.11. The sequence t n in (2.2) tends to 0 as ≈ 1/n. This is consistent with the theory of unavoidable sets (see Remark 2.2), where it is shown for that the minimal size of an unavoidable set (for the full shift on A) is ≫ |A| n /n ( [21,16]).

Symbolic model: large holes
The goal of this section is to extend Theorem 1.3 to more general subshifts.
Theorem 3.1. Let X be a subshift endowed with a mixing probability measure of maximal entropy µ. Let Y ⊂ X be a subshift such that 0 < h(Y ) < h(X) (i.e., a proper subshift of X).
Then for any ε > 0 there exists a finite overlapping union of cylinders G such that We see that Y ⊂ J (Σ ′ n ) and thus, h(J (Σ ′ n )) > 0. To get overlapping, let [w * ] ∈ Σ ′ n be such that µ([w * ]) is minimized. We note here that µ([w * ]) > 0 as µ is positive on all cylinders. Consider n (by mixing). That is, so that G n is overlapping.
Let m ≥ 2 and M be an m × m matrix with integer entries and determinant ±1. Then M determines the algebraic automorphism of the m-torus T m := R m /Z m , which we will denote by T M . That is, T M x = Mx mod Z m . We assume M to have a characteristic polynomial p irreducible over Q.
Assume that T M is hyperbolic, i.e., that p has no roots of modulus 1. Let t be a homoclinic point for T M , i.e., T n M t → 0 as n → ±∞. Let X be a two-sided subshift on a finite alphabet, and define the map φ t : X → T m as follows: where a = (a n ) n∈Z . These maps have been studied in [20,23,24,25]. Note first that φ t is well defined, since, as is well known, T n M t → 0 at an exponential rate, so this bi-infinite series converges. Also, φ t is Hölder continuous, for the same reason. Most importantly, we have φ t σ = T M φ t , i.e., φ t semiconjugates the shift and T M . It is known (see [20]) that one can choose L ≥ 1 large enough so that if X = {−L, . . . , L} Z is the full shift, then φ is surjective.
Assume now that p has one real root of modulus greater than 1 (β, say) and the rest are less than 1 in modulus. Then β is called a Pisot number (a Pisot unit, to be more precise, in view of det M = ±1) and T M a Pisot automorphism. In this case we have a natural choice for X, namely, X = X β , i.e., the natural extension of X + β endowed with the measure µ β , the natural extension of π −1 β (µ + β ) to X β . (This is the measure of maximal entropy for the subshift.) Let σ β : X β → X β denote the corresponding left shift. Note that since β is Pisot, (d ′ n ) ∞ 1 is eventually periodic and therefore, cannot contain unbounded strings of 0s (see, e.g., [4]).
As is well known, any homoclinic point t can be obtained by projecting a point in Z m onto the leaf of the unstable manifold for T M passing through 0 along the stable manifold. In the Pisot case this implies that T M t = βt, whence (4.3) can be written as (4.4) φ t (a) = n∈Z a n β −n t.
It has been shown independently in [20] and [23] that φ t is surjective and finite-to-one, i.e., there exists M ≥ 1 such that φ −1 t (x) is at most M points for any x ∈ T m . Furthermore, (X β , σ β , µ β ) is known to be irreducible sofic in this setting ( [4]).
Recall, we define distance on a subshift (X, µ, σ) with the alphabet A = {0, 1, . . . , k − 1} in the usual way as We denote distance on T m by |x − y|. We say that a map φ : X → T m is α-Hölder continuous if there exists a C > 0 such that for all x, y ∈ X we have

Auxiliary results.
Lemma 4.1. Let (X, µ, σ) be an irreducible sofic subshift endowed with the measure of maximal entropy, on the alphabet A of cardinality k. Assume we have an α-Hölder continuous map φ : X → T m such that φσ = T M φ.
Then there exist C, κ > 0 such that for any Borel set A ⊂ X we have Here   Proof. Since T M is continuous, we have that if φ t ([w]) is connected, then so is φ t (σ k ([w])) for all k ∈ Z, in view of φ t σ k β = T k M σ β . So, without loss of generality, we may prove the result for [w] only.
Recall that π β (X + β ) = [0, 1), a path connected set. This implies that for any cylinders The key observation to see this is, for w n = 0 that w 1 w 2 . . . w n−1 w n 000.... = w 1 w 2 . . . w n−1 (w n − 1)d 1 d 2 d 3 . . . and Now the claim follows from (4.4); indeed, for any two cylinders [w], [w ′ ] in X β , we can use the same chain as above (to be more precise, their two-sided analogues), and the images [w (j) ] and [w (j+1) ] under φ t will intersect as well. Proof. Let Ξ n denote the set of all cylinders [w −n+1 . . . w 0 .w 1 . . . w n ] in X β . Since φ t is surjective, there exists C = [w −n+1 . . . w n ] ∈ Ξ n such that φ t (C) has non-empty interior. Notice that as the word with all 0s has no constraints in X β as long as 2n ≥ ℓ, which we assume for now. This implies that which proves the claim, since any cylinder contains a longer one, hence 2n ≥ ℓ is not a real constraint.
Lemma 4.6. There exists c = c(M) > 0 such that for any cylinder C ∈ Ξ n we have that φ t (C) contains a cube whose sides are aligned with the axes, with side cβ −n .

4.3.
Toral automorphisms with small holes. Definition 4.7. Let X be a compact metric space, T : X → X be a continuous invertible map and µ be an ergodic T -invariant probability measure. We say that the dynamical system (X, µ, T ) possesses Property S if for any ε > 0 there exists an open connected subset A of X such that (1) µ(A) < ε; (2) for all, except, possibly, a countable set of x ∈ X, there exists n = n(x) ∈ Z, such that T n x ∈ A. Proof. Fix ε > 0 and choose C from Theorem 2.4 applied to X β . Put D = φ t (C). Clearly, D is compact; it is also path connected by Lemma 4.3. Furthermore, we have that φ t (µ β ) = H m , as a unique invariant measure of maximal entropy for T M . The set D is a complete trap; indeed, for x ∈ T m , let a = (a n ) ∞ −∞ ∈ φ −1 t (x). Since C is a complete trap, there exists n ∈ Z such that σ n a ∈ C. We then have T n M x = φ t (σ n a) ∈ D. By Corollary 4.2, H m (D) = O(ε κ ) with some κ > 0. Finally, for δ > 0 put D ′ = {x ∈ T m : |x − y| < δ, ∀y ∈ D}. Since D is path connected, we have that D ′ is an open (path) connected set containing D which can be made arbitrarily close to D in measure by choosing an arbitrarily small δ.  Proof. Suppose T = T M is Pisot. Since the orbit of T is the same as of T −1 for any x ∈ T m , it is clear from the definition that a complete trap for T is a complete trap for T −1 .
Not let S = −T and put E = D ′ ∪ (−D ′ ), where D ′ is constructed in the proof of Proposition 4.8. Clearly, E is open and has a small measure if D ′ does. Since S n = T n if n is even and −T n otherwise, we have that E is a complete trap for S if D ′ is such for T . To make E connected, we connect D ′ and −D ′ by an open "tunnel", which we can make as small in measure L m as we please. Let E ′ denote the resulting set, which is clearly small in measure and a complete trap for S. Now, the same E ′ works for the case U = −T −1 , which completes the proof.
Corollary 4.11. Any hyperbolic automorphism of T 2 or T 3 has Property S.
Proof. Clearly, any hyperbolic 2 × 2 matrix has one eigenvalue of modulus greater than one and one less than one. This makes it generalized Pisot. For a 3 × 3 hyperbolic matrix M we have that one of its eigenvalues is less than one in modulus and two greater than one or the other way round. In either case, T M is generalized Pisot so we can apply Theorem 4.10.
Consider now a general hyperbolic toral automorphism T M . Arithmetic symbolic models (similar to the one described above for the Pisot automorphisms) have been suggested by various authors -see [22,Section 4] for more detail. Unfortunately, none of these appears to produce an explicit symbolic coding space. Assume S = −T now. We will modify our proof of Theorem 3.1 in such a way that we will get H ⊂ T m with H = −H. Namely, put for any x ∈ X β , M(x) = {y ∈ X β : φ t (y) = ±φ t (x)}.
Since φ t is finite-to-one, there exists M ≥ 1 such that #M(x) ≤ M for all x ∈ X β . Now let Y be a subshift of X β from Theorem 3.1; for instance, we can take Y = X β ′ where 1 < β ′ < β. Put Roughly speaking, Y consists of all elements of Y together with their -possibly multiple -negatives. We claim that Y is shift-invariant (hence a subshift). Indeed, let y ∈ Y ; then we have φ t (y) = ±φ t (x) for some x ∈ Y . Thus, φ t (σ(y)) = T φ t (y) = ±T φ t (x) = ±φ t (σ(x)), whence σ(y) ∈ Y , since σ(x) ∈ Y . Furthermore, h( Y ) = h(Y ), since we only add at most M sequences for each x ∈ Y . By our construction, −φ t ( Y ) = φ t ( Y ). Following the proof of Theorem 3.1, we set Σ n := Σ n \ [w] : w ∈ L( Y ) .
Note that if K is even then S K = T K . To get connectedness, we again let [w * ] ∈ Σ n be such that µ[w * ] is minimized and let  To our best knowledge, this effect has not been studied in the literature (see Section 1).