Preamble

Functional Analysis and Its Applications, Vol. 2, No. 3, 1968, 70–80.

CONSTRUCTION OF MARKOV PARTITIONS

G. Sinai

In [1], the properties of U-diffeomorphisms were investigated using the Markov partitions introduced therein. The entire presentation in [1] was based on the fact that such partitions exist with elements of sufficiently small size. The present paper describes the construction of these Markov partitions, thereby demonstrating their existence. For terminology and notation, see [1].

Construction of a Markov Partition in the Two-Dimensional Case

We describe in detail the construction of the required partition in the two-dimensional case. Let $M$ be a two-dimensional torus and $T$ be its U-diffeomorphism. We shall call a curvilinear quadrilateral a "parallelogram" if one pair of its sides consists of segments of the expanding foliation and the other pair consists of segments of the contracting foliation. It is clear that within a parallelogram, any two segments of the same transversal foliation whose boundaries belong to the boundary of the parallelogram are canonically isomorphic. If $U$ is a parallelogram, then $\Gamma_p(U)$ denotes the part of the boundary $\partial U$ consisting of the two segments of the expanding foliation. $\Gamma_c(U)$ is defined analogously. We define $\text{diam}_p(U)$ ($\text{diam}_c(U)$) as the maximum length of a segment of the expanding (contracting) foliation inside $U$, and $\text{diam}(U) = \max(\text{diam}_p(U), \text{diam}_c(U))$.

Lemma 1.1. For any $\delta > 0$, the torus $M$ can be represented as a union of a finite number of parallelograms $U_1, \dots, U_s$ such that $M = \bigcup U_i$, $\text{int } U_i \cap \text{int } U_j = \emptyset$, and $\text{diam } U_j < \delta$ for $j = 1, \dots, s$.

Proof. Suppose that a finite number of segments of the contracting and expanding foliations are drawn on the torus such that the endpoint of each segment lies strictly inside a segment of the other foliation. If the sizes of the segments are sufficiently small, such a collection of segments will define a system of parallelograms of the required type: indeed, an open connected set bounded by such segments will be an open parallelogram. Furthermore, there exists a $\delta_1 > 0$ such that if the length of each segment does not exceed $\delta_1$, then the diameter of each parallelogram does not exceed $\delta$. Thus, our task is reduced to constructing a system of segments with the required properties.

Take a finite $\epsilon_1$-net. The requirements for the smallness of $\epsilon_1$ are specified below. Through each point of this net, draw a segment of the expanding foliation centered at that point with length $\delta$. By slightly shifting the points of our $\epsilon_1$-net so that they form a $2\epsilon_1$-net, we can ensure that the constructed segments do not intersect. Let $\epsilon_1$ be so small that the following property holds: if we draw a segment of the contracting foliation centered at an arbitrary point with length $\delta/4$, then each half of this segment will intersect at least once one of the expanding segments we constructed; moreover,

the distance of the intersection point from the midpoint of the expanding segment (i.e., from the point of our $2\epsilon_1$-net) does not exceed $\delta/4$.

Let us return to our situation. Let $\gamma_p$ be one of the expanding segments already constructed. Take its endpoint and draw a segment $\gamma_c$ of the contracting foliation of length $\delta/2$ centered at this endpoint. Then extend the segment $\gamma_c$ in both directions until it intersects an existing segment of the expanding foliation at a distance no greater than $\delta/4$ from its midpoint. The endpoints of the extended segment $\gamma_c$ will lie on the segments $\gamma_p$ of the expanding foliation, and their length will not exceed $\delta$ due to the choice of $\epsilon_1$. The lemma is proved.

Without fear of confusion, we shall call the constructed system of parallelograms a partition of the space $M$. For a partition $\alpha$, let $\Gamma_p(\alpha) = \bigcup \Gamma_p(U_i)$ and $\Gamma_c(\alpha) = \bigcup \Gamma_c(U_i)$.

Definition 1.1. A partition $\alpha$ is called Markov if for some integer $m > 0$:
$$T^m \Gamma_p(\alpha) \subset \Gamma_p(\alpha), \quad T^{-m} \Gamma_c(\alpha) \subset \Gamma_c(\alpha) \quad \text{(1)}$$

Starting from the partition $\alpha$ constructed in Lemma 1.1, we shall construct a Markov partition. First, we construct a partition $\alpha_1$ for which the first inclusion in (1) holds; in doing so, each segment in $\Gamma_c(\alpha)$ will either slightly increase or slightly decrease while remaining on its leaf.

Thus, we have a partition $\alpha$ into parallelograms $U_1, \dots, U_s$ constructed in Lemma 1.1, with $\text{diam}(U_j) \leq \delta$. Let us identify the points forming the $2\epsilon_1$-net that serve as centers of the expanding foliation segments of length $\delta$ (see Lemma 1.1). We denote these segments by $\Gamma_p^{(1)}, \dots, \Gamma_p^{(k)}$, where $k$ is the number of points in our $2\epsilon_1$-net. It is obvious that the endpoints of each segment $\Gamma_p^{(i)}$ belong to $\Gamma_c(\alpha)$ and that $\Gamma_p^{(i)}$ cannot be extended while maintaining this property. Furthermore, $\bigcup \Gamma_p^{(i)} = \Gamma_p(\alpha)$.

Since the pairwise distances between different $\Gamma_p^{(i)}$ are positive, there exists some $\epsilon_p(\alpha)$ such that each segment $\Gamma_p^{(i)}$ admits a continuous deformation during which it remains a segment of the expanding foliation with endpoints on $\Gamma_c(\alpha)$, staying within a distance $\epsilon_p(\alpha)$ of the original segment $\Gamma_p^{(i)}$ and at least $4\epsilon_p(\alpha)$ away from other segments $\Gamma_p^{(j)}$. Choose $m$ large enough so that $A_p(\alpha) \lambda_c^m < \epsilon_p(\alpha)$, where the constant $A_p(\alpha)$ will be specified shortly. Consider $T^{-m} \Gamma_p^{(i)}$; this is a segment of the expanding foliation of very small length. Find the point of the original $2\epsilon_1$-net closest to it and the corresponding segment $\Gamma_p^{(j)}$. By drawing segments of the contracting foliation through the endpoints of $T^{-m} \Gamma_p^{(i)}$ until they intersect $\Gamma_p^{(j)}$, we obtain a new segment $\tilde{\Gamma}_p^{(i)}$ on $\Gamma_p^{(j)}$. Let $A_p(\alpha)$ denote the maximum length of a segment of the contracting foliation connecting the points of $T^{-m} \Gamma_p^{(i)}$ with the corresponding points of $\Gamma_p^{(j)}$. One could define $A_p(\alpha)$ more formally, but the given description is likely sufficient. Consider $T^m \tilde{\Gamma}_p^{(i)} = \Gamma_{p,1}^{(i)}$. It is not difficult to see that the segment $\Gamma_{p,1}^{(i)}$ can be obtained from $\Gamma_p^{(i)}$ by the deformation described above that defines $\epsilon_p(\alpha)$. We denote the collection of all segments $\Gamma_{p,1}^{(i)}$ by $\Gamma_{p,1}$.

It is clear that the endpoint of each segment $\Gamma_{p,1}^{(i)}$ lies on some segment from $\Gamma_c(\alpha)$. However, the endpoints of the segments $\Gamma_c^{(j)}(\alpha)$ from $\Gamma_c(\alpha)$ do not necessarily lie on $\Gamma_{p,1}$. But it is easy to see that we can extend each segment from $\Gamma_c(\alpha)$ by no more than $A_p(\alpha) \lambda_c^m$ so that its endpoint falls on $\Gamma_{p,1}$. We denote the collection of segments modified in this way by $\Gamma_{c,1}$.

The system of segments $\Gamma_{p,1}$ and the system of segments $\Gamma_{c,1}$ possess the property specified at the beginning of the proof of Lemma 1.1: the endpoints of each segment of one foliation lie strictly inside a segment of the other foliation. Therefore, $\Gamma_{p,1}$ and $\Gamma_{c,1}$ generate a certain partition $\alpha_1$ into parallelograms, for which $\Gamma_{p,1} = \Gamma_p(\alpha_1)$, $\Gamma_{c,1} = \Gamma_c(\alpha_1)$, and $T^{-m} \Gamma_p(\alpha_1) \subset \Gamma_p(\alpha_1)$.

We perform a similar construction with $\alpha_1$. Note first that the magnitude of the shift of each $\Gamma_p^{(i)}$ does not exceed $A_p(\alpha) \lambda_c^m$. Since $T^{-m} \Gamma_{p,1} \subset \Gamma_{p,1}$, $T^{-m} \Gamma_{p,1}$ is separated in the direction of the contracting foliation from $\Gamma_{p,1}$ by no more than $A_p(\alpha) \lambda_c^m$. Map $T^{-m} \Gamma_{p,1}$ canonically onto $\Gamma_{p,1}$. The image of $T^{-m} \Gamma_{p,1}$ is separated in the direction of the contracting foliation from $T^{-m} \Gamma_{p,1}$ by no more than $A_p(\alpha) \lambda_c^m$. Then $T^m (T^{-m} \Gamma_{p,1}) = \Gamma_{p,2}$ is obtained from $\Gamma_{p,1}$ by the deformation described above by no more than $A_p(\alpha) \lambda_c^{2m}$. We denote the collection of segments $\Gamma_{p,2}^{(i)}$ by $\Gamma_{p,2}$. Extend each segment from $\Gamma_c(\alpha_1)$ by no more than $A_p(\alpha) \lambda_c^{2m}$ so that its endpoints are on $\Gamma_{p,2}$. We denote the collection of extended segments by $\Gamma_{c,2}$. Then $\Gamma_{p,2}$ and $\Gamma_{c,2}$ together generate a partition $\alpha_2$ for which $\Gamma_{p,2} = \Gamma_p(\alpha_2)$, $\Gamma_{c,2} = \Gamma_c(\alpha_2)$, and $T^{-m} \Gamma_p(\alpha_2) \subset \Gamma_p(\alpha_2)$.

Continuing this process, we obtain a sequence of partitions $\alpha_n$ for which:
1. $T^{-m} \Gamma_p(\alpha_n) \subset \Gamma_p(\alpha_{n-1})$;
2. Each segment from $\Gamma_p(\alpha_n)$ is canonically isomorphic to the corresponding segment from $\Gamma_p(\alpha_{n-1})$; the magnitude of the contracting foliation segment connecting corresponding points on $\Gamma_p(\alpha_n)$ and $\Gamma_p(\alpha_{n-1})$ does not exceed $A_p(\alpha) \lambda_c^{nm}$;
3. Each segment from $\Gamma_c(\alpha_n)$ is obtained by increasing or decreasing the corresponding segment from $\Gamma_c(\alpha_{n-1})$ by no more than $A_p(\alpha) \lambda_c^{nm}$.

If $m$ is sufficiently large, the sum of all shifts does not exceed $\epsilon_p(\alpha)$. Therefore, we can continue the process indefinitely and obtain in the limit a partition $\bar{\alpha}$. For it:
1. $T^{-m} \Gamma_p(\bar{\alpha}) \subset \Gamma_p(\bar{\alpha})$;
2. Each segment from $\Gamma_p(\bar{\alpha})$ is canonically isomorphic to the corresponding segment from $\Gamma_p(\alpha)$; the magnitude of any contracting segment effecting this isomorphism does not exceed $A_p(\alpha) \lambda_c^m / (1 - \lambda_c^m)$;
3. Each segment from $\Gamma_c(\bar{\alpha})$ is obtained from the corresponding segment from $\Gamma_c(\alpha)$ by changing its length by no more than $2 A_p(\alpha) \lambda_c^m / (1 - \lambda_c^m)$.

Additional Remark. Following the construction process of $\bar{\alpha}$, it is easy to verify that for each segment from $\Gamma_p(\bar{\alpha})$ canonically isomorphic to the corresponding $\Gamma_p^{(i)} \subset \Gamma_p(\alpha)$, we have $T^{-m} \Gamma_p^{(i)} \subset \Gamma_p^{(i)}$, and the distance from $T^{-m} \Gamma_p^{(i)}$ to the boundary of $\Gamma_p^{(i)}$ does not exceed $\delta/2$. From this follows the next fact: take an arbitrary partition $\alpha'$ for which the segments $\Gamma_p^{(i)}$ from $\Gamma_p(\alpha')$ are obtained only by changing the length of each $\Gamma_p^{(i)}$ by no more than $\delta/8$ while remaining on its leaf. Then, if $m$ is so large that $A_p(\alpha) \lambda_c^m < \delta/8$, then $T^{-m} \Gamma_p(\alpha') \subset \Gamma_p(\alpha')$.

Now apply the process described above to $\bar{\alpha}$, replacing $T$ with $T^{-1}$. Since in this case each segment $\Gamma_p^{(i)}$ changes as described in the additional remark regarding the transition from $\alpha$ to $\alpha'$, the property $T^{-m} \Gamma_p(\bar{\alpha}) \subset \Gamma_p(\bar{\alpha})$ will be consistently preserved during our process. For the partition $\alpha$ obtained in the limit, $T^{-m} \Gamma_p(\alpha) \subset \Gamma_p(\alpha)$ and $T^m \Gamma_c(\alpha) \subset \Gamma_c(\alpha)$. Consequently, $\alpha$ is a Markov partition.

§ 2. Construction of a Markov partition

General Case

The idea of constructing a Markov partition was demonstrated in the previous section using the example of a two-dimensional torus. Here, we examine the general case. Conceptually, it does not differ from the two-dimensional case. This section may be skipped upon a first reading. Let $U$ be an arbitrary set contained within a sufficiently small spherical neighborhood $O$. We define a local fiber (l.f.) of any of the foliations in $U$ as the intersection of $U$ with a complete local fiber in $O$; this intersection may be disconnected. Henceforth, unless otherwise specified, we refer only to local fibers within some set $U$, which will be clear from the context.

Definition 2.1. A set $U$ is called a parallelogram if both the expanding local fibers (e.l.f.) and the contracting local fibers (c.l.f.) in $U$ are canonically isomorphic to each other. A parallelogram may also be disconnected. Let us take a point $x \in U$ and the e.l.f. $D_p(x)$ and c.l.f. $D_c(x)$ passing through $x$.

For each $z \in D_p(x)$, let $D_c(z)$ denote the local fiber canonically isomorphic to $D_c(x)$. Then $D_c(z) \subset U$. Consequently, $\bigcup_{z \in D_p(x)} D_c(z) \subset U$. We shall show that $U = \bigcup_{z \in D_p(x)} D_c(z)$. Indeed, for every $y \in U$, the c.l.f. $D_c(y)$ in $U$ passing through $y$ is canonically isomorphic to $D_c(x)$.

Consider the point $z \in D_p(x)$ that is canonically isomorphic to $y$. It is clear that $z \in D_p(x)$, and then $D_c(y) = D_c(z)$, which was to be shown. These arguments demonstrate, first, that each parallelogram is uniquely determined by its fibers $D_p$ and $D_c$ for any point $x \in U$, and second, they provide a method for constructing parallelograms. Specifically, by taking an arbitrary point $x_0$ and the local fibers $D_c(x_0)$ and $D_p(x_0)$ passing through $x_0$, we set $U = \bigcup_{z \in D_p(x_0)} D_c(z)$, where each $D_c(z)$ is canonically isomorphic to $D_c(x_0)$. The closure of a parallelogram is again a parallelogram, $\bar{U} = \bigcup_{z \in \bar{D}_p(x_0)} \bar{D}_c(z)$.

Lemma 2.1. The intersection $U_1 \cap U_2$ of two parallelograms $U_1$ and $U_2$ is a parallelogram.

Proof. Let $x \in U_1 \cap U_2$, and let $D_p^{(1)}(x)$ and $D_p^{(2)}(x)$ be the e.l.f. in $U_1$ and $U_2$ respectively, and similarly $D_c^{(1)}(x)$ and $D_c^{(2)}(x)$ for the c.l.f. We set...

Dp{x) = Di'\x)nDi'\x\

Dc{x)^D^^\x)f]Di'\x),

Ya. G. Sinai. It is easily verified that $U[f]U^{-1}$ is canonically isomorphic to $D_C(A')$. The lemma is proven. Let $U$ be a parallelogram, and let $D_P(x)$ and $D_C(x)$ be the local manifolds (l.m.) defining it. The boundary is given by $\partial U = \bigcup D_C(x) \cup \bigcup D_P(x)$, where $\Gamma_C(U) = \bigcup D_C(x)$ and $\Gamma_P(U) = \bigcup D_P(x)$ (the bar denotes closure). In accordance with the convention adopted in \cite{1}, the local manifolds ($D_P(x)$) are assumed to be admissible; that is, they possess an open interior whose boundary has measure zero. We shall define a partition into parallelograms as a system of parallelograms $\{U_i\}$ such that $M = \bigcup U_i$ and $U_i \cap U_j = \emptyset$ for $i \neq j$ (specifically, if the sets are open, then $U_i \cap U_j = \emptyset$). Let $\xi$ be a partition into parallelograms. We define $\Gamma_C(\xi) = \bigcup \Gamma_C(U_i)$ and $\Gamma_P(\xi) = \bigcup \Gamma_P(U_i)$.

Definition 2.2. A partition $\xi$ is called a Markov partition if, for some $k$, $T(\Gamma_C(\xi)) \subset \Gamma_C(\xi)$ and $T^{-1}(\Gamma_P(\xi)) \subset \Gamma_P(\xi)$.

Suppose we are given an arbitrary cover consisting of a finite system of open parallelograms of sufficiently small diameter. We shall denote this cover by $\{G_i\}$. Given the cover $\{G_i\}$, we introduce a metric in the space of covers $\{G'_i\}$ that differ only slightly from $\{G_i\}$. Assume that $G_i \cap G'_i \neq \emptyset$ for $x_i \in G_i \cap G'_i$. Let $G_i$ be defined by the expanding local manifold (e.l.m.) $D_P(x_i)$ and the contracting local manifold (c.l.m.) $D_C(x_i)$, while $G'_i$ is defined by $D'_P(x_i)$ and $D'_C(x_i)$. We define the distance between $G_i$ and $G'_i$ as the maximum of the distances $d_P(D_P(x_i), D'_P(x_i))$ and $d_C(D_C(x_i), D'_C(x_i))$. We define the distance between the covers $\{G_i\}$ and $\{G'_i\}$ as the maximum of the distances between $G_i$ and $G'_i$. We denote this distance by $d(\{G_i\}, \{G'_i\})$. (Naturally, our metric depends on the choice of points $x_i$. In our subsequent constructions, it will be clear which points are intended.) Having an open cover $\{G_i\}$, we now construct a partition into parallelograms from it. We choose points $x_i$ and introduce the c.l.m. and e.l.m.

$D_P(x_i)$ that define the parallelogram $G_i$. For each $i$, we consider those $j$ for which $G_i \cap G_j \neq \emptyset$. Let $G_{ij}^{(C)} = G_i \cap G_j$. It is clear that $G_{ij}^{(P)}$ and $G_{ij}^{(C)}$ are parallelograms and that the parallelograms $G_i \cap G_j = G_{ij}^{(C)} \cap G_{ij}^{(P)}$, $G_{ij}^{(C)}$, and $G_{ij}^{(P)}$ are uniquely determined by open admissible subsets $D_{ij}^{(P)} \subset D_P(x_i)$ and $D_{ij}^{(C)} \subset D_C(x_i)$. The subsets $D_{ij}^{(C)}$, to which $D_C(x_i)$ is attached, form an open cover of $D_C(x_i)$. Denoting the elements of this cover by $E_{i1}(c), E_{i2}(c), \dots$, we set $E_{ij}^{(1)} = E_{ij}(c)$ and $E_{ij}^{(-1)} = D_C(x_i) \setminus E_{ij}(c)$, and form a partition $\phi_i^{(C)}$ of the c.l.m.

$D_C(x_i)$ produced by the intersections $\bigcap E_{ij}^{(\epsilon_j)}$ for all possible sequences $\epsilon_2, \epsilon_3, \dots$ taking values $\pm 1$.

Construction of Markov Partitions

We denote the elements of the resulting partition by $F_{ik}(c)$. Since the sets $E_{ij}$ are either open or closed, each element is the intersection of open and closed sets. The boundary of this closed set has measure zero. Henceforth, we consider only those $F_{ik}(c)$ for which the interior (i.e., the open kernel) is non-empty; the others are closed sets of measure zero that are contained within the boundaries of the former. We now construct an analogous partition $\phi_i^{(P)}$ of the e.l.m.

$D_P(x_i)$ into sets $F_{il}(p)$ and set

$H_{ij,k} = \bigcup F_p(z) = \bigcup F_c(z)$

$F_p(z) \cap F_c(z)$ is a point, canonically isomorphic to $F_{\alpha^k(p)} \cap F_{\omega^k(c)}$. Clearly, the parallelograms form a partition of each $G_i$. Moreover, any parallelogram $G_{j_0} \cap \dots \cap G_{j_s}$ for any $s$ can be represented as a sum of parallelograms $H_{ij,k}$. Indeed, the local fibers forming it are sums of elements from the partitions $\Phi_p$ and $\Phi_c$. It follows from this statement that every set $G_{j_0 \dots j_s} = G_{j_0} \cap \dots \cap G_{j_s} \setminus \bigcup G_l$ is also partitioned by the parallelograms $H_{ij,k}$. While the sets $G_{j_0 \dots j_s}$ are not themselves parallelograms, they form a partition (in the same sense as we defined the partition into parallelograms). Furthermore, each $G_{j_0 \dots j_s}$ is partitioned in $s+1$ ways by the parallelograms $H_{ij,k}$, where $k = 0, 1, \dots, s$. The partition of the set $G_{j_0 \dots j_s}$ produced by all possible intersections $H_{j_0(0), (0)} \cap H_{j_1(1), (1)} \cap \dots \cap H_{j_s(s), (s)}$ no longer depends on $k$ and forms a partition of $G_{j_0 \dots j_s}$ into parallelograms. Since the distinct $G_{j_0 \dots j_s}$ are pairwise disjoint, we obtain a partition of the entire space into parallelograms. This is the required partition, which we denote by $\eta$. We now examine the boundary of the partition $\eta$. We restrict our consideration to $\Gamma_p$, as $\Gamma_c$ is treated analogously. Since for any two partitions $\xi_1$ and $\xi_2$ into parallelograms, $\Gamma_p(\xi_1 \cdot \xi_2) \subset \Gamma_p(\xi_1) \cup \Gamma_p(\xi_2)^*$, it follows that $\Gamma_p(\eta) = \bigcup \Gamma_p(H_{ij,k})$. For every parallelogram $H_{ij,k}$, its boundary is $\Gamma(H_{ij,k}) = \Gamma_p(H_{ij,k}) \cup \Gamma_c(H_{ij,k})$, where $\Gamma_p(H_{ij,k}) = \bigcup_{z \in \partial F_{\alpha^k(p)}} F_c(z)$ and $\Gamma_c(H_{ij,k}) = \bigcup_{z \in \partial F_{\omega^k(c)}} F_p(z)$. From this relation and the construction, it is evident that if a piecewise linear fiber (p.l.f.)

$F_p(z) \in \Gamma_p^{(i,k)}$, then the p.l.f. $D_p(z) \subset G_i$ containing it consists entirely of local fibers belonging to the boundaries of other $H_{ij,k}$. Thus, inside $G_i$,

$\Gamma_p^{(i,k)} \cap \text{int}(G_i) = \bigcup \Gamma_p^{(j,m)} = \bigcup \bigcup D_p(z)$

We further note that $D_p(z)$ is contained in the last sum if and only if $D_p(z)$ intersects at least one p.l.f. $D'(z) \in \Gamma_p(G_j)$ for some $j$. This p.l.f.

$D_p(z)$ is included in $\Gamma_p(\eta)$ by construction. It follows that $\Gamma_p(G_j) \subset \bigcup \Gamma_p^{(i,k)} = \Gamma_p(\eta)$.
*) By $\xi_1 \cdot \xi_2$ we mean, as usual, the partition produced by the pairwise intersections of elements of one partition with elements of the other.

Ya. G. Sinai. Let us take $D_p(z) \subset T_p(G_j)$, $\int dD_c(x_j)$, and consider those $t$ for which $D_p(z) \neq 0$. Then there exists a local stable manifold (l.s.m.) such that $D_p(z) \subset D_p(z)$. We construct this for each l.s.m.

$D_p(z) \cup \dots \cup \dots$. It follows from the preceding that the entire l.s.m.

$D_p(z)$, such that the sum of all such $D_p(z)$ yields the entire boundary $T_p(\eta)$. If the indices are the same for two points, then $D_p(z_1)$ and $D_p(z_2)$ are canonically isomorphic. We define a boundary component $\Gamma_p(\eta)$ as the collection of all canonically isomorphic l.s.m. that share the same indices. We shall denote these components by $\Gamma_i$. It is clear that $\bigcup \Gamma_i = T_p(\eta)$. We now proceed directly to the construction of the Markov partition. First, we construct a covering by parallelograms. Let us choose sufficiently small numbers $\delta$ and $\epsilon$. The requirements for their smallness will be specified below. Take a finite $\delta$-net $x_1, \dots, x_n$ and open balls $B(x_i; \epsilon/2)$ of radius $\epsilon/2$. We construct parallelograms $D(z) = D_c(z) \times D_p(z)$, where $D_c(z)$ is canonically isomorphic to $D_c(x_i; \epsilon/2)$. Our first requirement for $\delta$ and $\epsilon$ is that these parallelograms form a covering. For reasons that will become apparent later, it is expedient to expand these and, retaining the notation, pass to $G_i = D_c(z) = \bigcup \dots$. These new $G_i$ certainly form a covering. We construct a partition based on them. Consider an arbitrary full leaf of the contracting foliation. The connected components of the intersection of $\Gamma^{(s)}$ with $G_i$ are, in fact, nothing other than certain l.s.m.

$D_c(z) \subset G_i$, which form an open covering of the leaf $\Gamma^{(s)}$. Since the l.s.m. lying within the original parallelograms constructed from balls of radius $\epsilon/2$ also form a covering, there exists a fixed number $d > 0$ with the following property: if we take open $D_c(z) \subset \Gamma^{(s)}$ such that $\min d_p(x, y) \geq d$, then such l.s.m. also form an open covering of $\Gamma^{(s)}$. Any l.s.m. for which this last inequality holds will be called "fenced." Let $U$ be an arbitrary bounded subset of $\Gamma^{(s)}$. The set of all fenced l.s.m. intersecting $U$ will be called the "section containing $U$" and denoted by $V(U)$. If $\Lambda_c$ is the maximum diameter of $D_c(z)$ (over all $z \in \dots$), then the distance (in the metric of the contracting leaf) from the boundary of the section $V(U)$ to $U$ does not exceed $\Lambda_c$. The section is an open subset of $\Gamma^{(s)}$. Obviously, it is the union of a finite number of l.s.m.

$D_c(z)$. If a fenced l.s.m. $D_c(z) \subset V(U)$ but is not entirely contained in $U$, we shall call it a "boundary" l.s.m. We define the "fence" of the section $V(U)$, denoted by $\partial V(U)$, as $\bigcup D_c$, where the union is taken over the boundary fenced l.s.m. $D_c$. Since for some $i$, where the partition is constructed from $\{G_i\}$, it follows that $\partial V(U) \subset \Gamma_p(\eta)$ for every $U$. The objects $V(U)$, $\partial V(U)$, and the number $d$ depend continuously on the covering $\{G_i\}$. We will need the following special case of this statement: the set of coverings $\{G_i\}$ for which $d(\{G_i\}, \{G'_i\}) < \epsilon_1$, where $\epsilon_1$ is sufficiently small, and $D_p(x_i) \subset D_p(x_i)$, $D_c(z) \subset D_c(z)$ for every $z \in D_p(x_i; \delta)$ (and thus $G_i \supset G'_i$), possesses the following properties:

Construction of Markov Partitions: 1) The natural correspondence $D_c(z) \leftrightarrow D'_c(z)$ associates every fenced l.s.m.

$D_c(z)$ is a local stable manifold (l.s.m.) such that the height of the boundary $d_p(x, y) > d/2$, and $D_p(x)$ is a local unstable manifold (l.u.m.) lying in the same parallelogram as $D_c(z)$; such an l.s.m. in the covering parallelogram $\{G_i\}$ will also be called "fenced." 2) There exists such a fenced l.s.m. and an l.s.m. $\mu(D_c)$ canonically isomorphic to it, where the distance in the metric of the expanding layer between corresponding points exceeds $n(D_c)$ for the l.s.m. in the same $G_i$. 3) There exists a constant $A_p$ such that for every $D_p(z) \subset \Gamma_p(\eta)$, we have $\text{diam } D_p(z) < A_p$ and $\text{diam } D_p(z) < A_p$. If the covering $\{G_i\} \in \mathcal{O}_{\delta}$, then for any bounded set in any layer, we define it as the sum of those l.s.m. $D_c$ that correspond to (1) l.s.m. $D_c$, the sum of which yields the set. Similarly, $(\eta)_c \subset \Gamma_p(\eta')$ is defined. Now let $n$ be sufficiently large such that: 1) $A_c \lambda_c^n / (1 - \lambda_c^n) < \delta_1$, $A_p \lambda_p^n / (1 - \lambda_p^n) < \delta_1$; 2) $A_p \lambda_p^{-n} < \delta_n$, $A_c \lambda_c^n < \delta_2$; 3) $A_c \lambda_c^n < \delta_3$, $A_p \lambda_p^n < \delta_3$, where the specific values will be indicated later. Let $T^{-n} D_c(x_i) \in Y(T^{-n} D_c(x_i), \delta)$ and set $D_c^{(1)}(x_i) = T^n(Y(T^{-n} D_c(x_i), \delta))$. We construct $D_c^{(1)}(x_i)$, $D_p^{(1)}(x_i)$, and the parallelograms for the covering $\{G_p\}$ and the partition. Then $d(\{G_i\}, \{G_i^{(s+1)}\}) < A_c \lambda_c^n < \delta_1$. Therefore, $\{G_i^{(1)}\} \in \mathcal{O}_{\delta_1}$ and $\text{diam}_p D_p^{(1)}(z) < A_p$. By construction, $T^{-n} D_p(z) \subset D_p(T^{-n} z) \in \Gamma_p(G_i)$ for some $i$; moreover, $T^{-n} D_c^{(1)}(z)$ is separated from the boundary $\partial D_p(T^{-n} z)$ by less than $A_p \lambda_p^{-n}$, since $D_p(T^{-n} z)$ contains a ball of radius $d/2$ (see the property in the definition) and the set $T^{-n} D_c^{(1)}$ contains the center of this ball. We define the subsequent process inductively. Suppose the coverings $\{G_i^{(k)}\} \in \mathcal{O}_{\delta_1}$ have been constructed for $k = 1, \dots, s$, where: 1) the parallelogram is defined by the l.s.m.

$D_c^{(k)}(x_i)$ and the l.u.m. $D_p^{(k)}(x_i)$, such that $D_c^{(k)}(x_i) \subset D_c^{(k+1)}(x_i)$, and thus $G_i^{(k)} \subset G_i^{(k+1)} \subset \dots \subset G_i^{(s)}$. 2) If $D_c(z)$ is an l.s.m. in $z \in D_p(x_i)$, then $d_c(\partial D_c^{(k)}(z), \partial D_c^{(k-1)}(z)) < A_c \lambda_c^{kn}$. 3) If $Y_k$ is a region constructed using the covering $\{G_i^{(k)}\}$ and the corresponding partition, then $D_c^{(k+1)}(x_i) = T^n Y_k(T^{-n} D_c^{(k)}(x_i))$. Properties 1) and 3) define the transition from $\{G_i^{(k)}\}$ to $\{G_i^{(k+1)}\}$. It remains only to verify that property 2) is preserved. Let $D_c^{(k+1)}(z)$ be an l.s.m. canonically isomorphic to $D_c^{(k+1)}(x_i)$. Then $T^{-n}(D_c^{(k+1)}(z))$ is canonically isomorphic to $T^{-n}(D_c^{(k+1)}(x_i)) = Y_k(T^{-n} D_c^{(k)}(x_i))$. By construction, $Y_k(T^{-n} D_c^{(k)}(x_i))$ is the set-theoretic sum of fenced l.s.m. Since $A_p \lambda_p^{-n} < \delta_3$, the image of $D_c^{(k)}$ under this isomorphism will be some l.s.m. Consequently, $T^{-n}(D_c^{(k+1)}(z))$ is a sum of l.s.m. Furthermore, $T^{-n}(D_c^{(k)}(x_i)) = Y_{k-1}(T^{-n} D_c^{(k-1)}(x_i)) = \bigcup D_c^{(k-1)} \subset T^{-n} D_c^{(k+1)}(x_i) = \bigcup D_c^{(k)}$. Each $D_c^{(k-1)}$ from the first sum is contained in exactly one $D_c^{(k)}$ from the second.

From this and from (2), it follows that $d_c(\partial (\bigcup D_c^{(k)}), \partial (\bigcup D_c^{(k-1)})) < A_c \lambda_c^{kn}$ and, consequently, $d_c(T^n(\bigcup D_c^{(k)}), T^n(\bigcup D_c^{(k-1)})) < A_c \lambda_c^{(k+1)n}$. Furthermore, the canonical isomorphism maps $D_c^{(k+1)}(z)$ to $D_c^{(k+1)}(x_i)$ and $D_c^{(k)}(z)$ to $D_c^{(k)}(x_i)$. Since the isomorphism commutes with the shift, we have $d_c(\partial T^{-n} D_c^{(k+1)}(z), T^{-n} D_c^{(k)}(z)) < A_c \lambda_c^{kn}$. Therefore, $d_c(D_c^{(k+1)}(z), D_c^{(k)}(z)) < A_c \lambda_c^{(k+1)n}$. Thus, 2) is proved. From 3) it follows that if $\Gamma_p^s(\eta_s)$ is a component of the partition boundary $\eta_s$ defined by the subset $\partial D_c^{(s)}(x_j)$, then $T^{-n} \Gamma_p(\eta_s) \subset \partial Y_{s-1}(T^{-n} D_c^{(s)}(x_i)) \subset \Gamma_p(\eta_{s-1})$, and thus $T^{-sn} \Gamma_p(\eta_s) \subset \Gamma_p(\eta_{s-1})$. Let $D_c^{(\infty)}(x_i) = \bigcup D_c^{(k)}(x_i)$. Then the covering $\{G_i^{(\infty)}\} \in \mathcal{O}_{\delta_n}$ and for it $T^{-n} \Gamma_p(\eta_\infty) \subset \Gamma_p(\eta_\infty)$. Additional remarks:

If a region is defined by the covering $\{G_i^{(n)}\}$, then for any $D_p^{(n)}(z) \subset \Gamma_p(z)$, the image $T^{-n} D_p^{(n)}(z)$ is separated from the boundary of the local stable manifold (l.s.m.) $D_p^{(0)}(T^{-n} z) \subset \Gamma_p(T^{-n} z)$ containing it by a distance of at least $A_p \lambda^n$. Indeed, $D_p^{(n)}(T^{-n} z)$ is contained within some "fence" $B_n$. Therefore, the distance to the boundary of the point $d D_p^{(n)}(T^{-n} z)$ is at least $d/2$, while the diameter of $T^{-n} (D_p^{(n)})$ does not exceed $A_p \lambda^n$, given the properties of $\{G_i^{(n)}\}$. As previously demonstrated, if a local stable manifold lies within one of the parallelograms $G_i^{(n)}$, then $T^{-n} D_i^{(n)}$ is a union of local stable manifolds.

From this, it follows that $A_i^{(n)}(z) = T^{-n}(D_i^{(n)}(z_i))$ is a union of local stable manifolds $D_j^{(n-k)}$; similarly, $A_i^{(n)}(z) - T^{-n}(D_i^{(n)}(z_i))$ is also a union of local stable manifolds. The set $B_s(z) \setminus \{D_i^{(n)}(z_i)\}$ is a union of local stable manifolds, as $A_i^{(n-k)}$ is a sum of local stable manifolds. Each $D_c^{(n-k)}$ contains exactly one local stable manifold.

Regarding the $D_i^{(n-k)}$ contained within, it follows that $T^n B_s$ is separated from the boundary by no more than $A_c \lambda^n$. We now use this final observation to demonstrate that the limiting layers $D_i^{(n)}(z)$ are admissible. For each $D_i^{(n)}(z)$, we construct a decreasing sequence of sets $W_1(z) \supset W_2(z) \supset \dots$ such that $\bigcap W_s(z) = D_i^{(n)}(z)$ and $\sigma_c(W_{s+1}) \leq \rho \sigma_c(W_s)$ for some constant $\rho < 1$ and $s = 1, 2, \dots$. The required assertion will clearly follow from this.

Let $\Gamma^{(s)}$ be the full leaf of the contracting foliation containing the point $T^{-s} z$. We define $G_{i_1 \dots i_s} = G_{i_1} \cap \dots \cap G_{i_s}$. The sets $G_{i_1 \dots i_s}$ form a partition and, consequently, a partition of each $\Gamma^{(s)}$. We may assume from the outset that every non-empty $G_{i_1 \dots i_s}$ has an open kernel in whose closure it is contained; if this is not the case, we can achieve it by an arbitrarily small perturbation of the points.### Construction of Markov Partitions

The intersection of the full leaf $\Gamma^{(s)}$ with the sets $G_{i_1 \dots i_s}$ generates admissible local manifolds that form a partition, which we denote by $\tau(\Gamma^{(s)})$. There exists some $\epsilon > 0$ such that every element of the partition $\tau(\Gamma^{(s)})$ contains a ball of radius $\epsilon$. Since the diameter of each element of $\tau(\Gamma^{(s)})$ does not exceed $A_c$, then for any $\delta < \epsilon/2$, if $U_\delta(C)$ is the $\delta$-neighborhood of the boundary of $C$ (inside $C$), then $\frac{\sigma_c(U_\delta(C))}{\sigma_c(C)} < \delta$. We fix some $\delta$ and find the corresponding $U_\delta$. For any $s$, we introduce the set $W_s$ equal to the sum of the elements of the partition $\tau(\Gamma^{(s)})$ that are at a distance of no more than $2 A_c$ from the boundary of $B_s$. Then $d(T^n B_s(z))$ is contained in a $2 A_c \lambda^n$-neighborhood of the boundary $\partial B_{s-1}(z)$. Since, according to the remark, $d A_i^{(n)}$ is contained in an $A_c \lambda^n$-neighborhood of the boundary $d B_{s-1}(z)$, and $B_{s-1}$ consists of elements of the partition $\tau(\Gamma_{s-1})$, it follows that $W_s$ is contained in a $3 A_c \lambda^n$-neighborhood of $d B_{s-1}(z)$ and, consequently, in $W_{s-1}$. Let $W_s = T^{-n} W_s$. From the preceding arguments, it follows that... We now show that $\sigma_c(W_{s+1}) \leq \rho \sigma_c(W_s)$. Let $C_j^{(s)}$ be the elements of the partition $\tau(\Gamma^{(s)})$ that compose $W_s$. Then $W_s = \bigcup T^{-n}(C_j^{(s)} \setminus U_\delta(C_j^{(s)}))$. Consider the Jacobian $A_c(x)$ of the volume transformation on the leaf $\Gamma^{(s)}(x)$ passing through $x$ to the volume on the leaf $\Gamma^{(s)}(Tx)$. The function $A_c(x)$ satisfies a Lipschitz condition. We show that the product $A_c(x) \dots A_c(T^{n-1} x)$ depends weakly on $x \in C_j^{(s)}$. Specifically, $d_c(T^k x, T^k x'') \leq A_c \lambda^k$. Consequently, the ratio of the products is bounded by $e^L$, where $L$ is the Lipschitz constant. If $A_c$ is sufficiently small (which can be achieved by the smallness of $\delta$), the final product will be less than $1/2$. Consequently,
$$\frac{\sigma_c(T^{-n}(U_\delta(C_j^{(s)})))}{\sigma_c(T^{-n} C_j^{(s)})} \leq \frac{1}{4} \frac{\sigma_c(U_\delta(C_j^{(s)}))}{\sigma_c(C_j^{(s)})} < \frac{2\delta}{4} < 1.$$

Ya. G. Sinai. Finally, we have

$$ \hat{\zeta}_0 = \rho_{\gamma} $$

which was to be demonstrated. We must now apply a process analogous to the one described above, replacing the expanding foliation with the contracting one, in order to obtain the necessary inclusions for $\gamma_c(\alpha)$ while preserving the previously obtained inclusion for $\gamma_p(\alpha)$. Note that during the sequential reconstructions described, as we transitioned from $\eta_s$ to $\eta_{s+1}$, all stable local manifolds (s.l.m.) changed only slightly near their boundaries, remaining on the same contracting leaf. As we now begin the reconstruction, moving from $\eta_{\infty, 0} = \eta_0$ to $\eta_{\infty, 1}$, and so on, from $\eta_{\infty, s}$ to $\eta_{\infty, s+1}$, the unstable local manifolds (u.l.m.) will begin to change. However, the fundamental relation (2) will be preserved. Indeed, it follows from Remark 1 that for each u.l.m. $D_p^{(\infty, s)}(z) \in \eta_{\infty, s}$, its image $T^{-k} D_p^{(\infty, s)}(z)$ is separated from the boundary of the element containing it, $(z') \in \Gamma_p(\eta_{\infty, s})$, by a distance of at least $A_p \lambda^m$. For each covering $\{G_i^{(\infty, s+1)}\}$ that now arises, every u.l.m. $D_p^{(\infty, s+1)}(z)$ belonging to any of the parallelograms $G_i^{(\infty, s+1)}$ contains a u.l.m.

$D_p^{(\infty, s)}(z)$ from $G_i^{(\infty, s)}$, and the distance from $D_p^{(\infty, s)}(z)$ to the boundary of $D_p^{(\infty, s+1)}(z)$ does not exceed $A_p \lambda^{m+s}$. It follows from this that for all $\eta_{\infty, s}$, relation (2) holds. If we pass to the limit as $s \to \infty$, then for the limiting partition $\eta_{\infty, \infty}$, it follows that this partition will be Markovian. The fact that the local manifolds of the partition $\eta_{\infty, \infty}$ are admissible is proved similarly.

Moscow State University
Received by the Editorial Board
November 3, 1967

REFERENCES

Markov Partitions and $Y$-diffeomorphisms

Ya. G. Sinai

Funktsional'nyi Analiz i ego Prilozheniya, Vol. 2, No. 1 (1968), pp. 64–89.

Introduction

In the theory of dynamical systems, the study of $Y$-systems (Anosov systems) occupies a central position. These systems are characterized by a strong form of hyperbolic behavior, where the tangent space at each point splits into expanding and contracting subspaces. A fundamental problem in the ergodic theory of such systems is the construction of invariant measures and the analysis of their statistical properties, such as ergodicity, mixing, and the Bernoulli property.

The present paper introduces the technique of Markov partitions, which provides a powerful tool for representing the continuous dynamics of $Y$-diffeomorphisms as symbolic dynamics. By partitioning the manifold into specific sets that respect the local stable and unstable manifold structure, we can map the action of the diffeomorphism to a topological Markov chain (a shift of finite type). This representation allows us to apply the well-developed methods of statistical mechanics and information theory to the study of smooth dynamical systems.

1. Basic Definitions and Notations

Let $M$ be a smooth compact Riemannian manifold and $f: M \to M$ be a $Y$-diffeomorphism. For any point $x \in M$, there exists a decomposition of the tangent space $T_x M = E^s_x \oplus E^u_x$ such that for some $C > 0$ and $0 < \lambda < 1$:
1. $\|df^n v\| \le C \lambda^n \|v\|$ for $v \in E^s_x, n \ge 0$;
2. $\|df^{-n} v\| \le C \lambda^n \|v\|$ for $v \in E^u_x, n \ge 0$.

The distributions $E^s$ and $E^u$ are integrable, giving rise to global stable and unstable manifolds, denoted by $W^s(x)$ and $W^u(x)$, respectively. We denote local stable and unstable manifolds of size $\epsilon$ as $W^s_\epsilon(x)$ and $W^u_\epsilon(x)$.

2. Local Structure and Rectangles

A key concept in the construction