Markov partition

Markov partition is a fundamental concept in the mathematical theory of dynamical systems which allows one to represent a discrete dynamical system as a shift of finite type on an auxiliary space of sequences of abstract symbols. Such a partition shows that, at a coarse level, the deterministic dynamic system resembles a discrete-time Markov process and allows to apply methods of symbolic dynamics to the study of long-term dynamical characteristics of the system, such as its topological entropy.

Motivation

Let (M,φ) be a discrete dynamical system. A basic method of studying its dynamics is to find a symbolic representation: a faithful encoding of the points of M by sequences of symbols such that the map φ becomes the shift map.

Suppose that M has been divided into a number of pieces E₁,E₂,…,E_r, which are thought to be as small and localized, with virtually no overlaps. The behavior of a point x under the iterates of φ can be tracked by recording, for each n, the part E_i which contains φⁿ(x). This results in an infinite sequence on the alphabet {1,2,…r} which encodes the point. In general, this encoding may be imprecise (the same sequence may represent many different points) and the set of sequences which arise in this way may be difficult to describe. Under certain conditions, which are made explicit in the rigorous definition of a Markov partition, the assignment of the sequence to a point of M becomes an almost one-to-one map whose image is a symbolic dynamical system of a special kind called a shift of finite type. In this case, the symbolic representation is a powerful tool for investigating the properties of the dynamical system (M,φ).

Formal definition

A Markov partition^[1] is a finite cover of the invariant set of the manifold by a set E of curvilinear rectangles $E=\{E_{1},E_{2},\cdots E_{r}\}$ such that

For any pair of points $x,y\in E_{i}$ , that $W_{s}(x)\cap W_{u}(y)\in E_{i}$

$\operatorname {Int} E_{i}\cap \operatorname {Int} E_{j}=\emptyset$ for $i\neq j$

If $x\in \operatorname {Int} E_{i}$ and $\phi (x)\in \operatorname {Int} E_{j}$ , then

\phi \left[W_{u}(x)\cap E_{i}\right]\supset W_{u}(\phi x)\cap E_{j}

\phi \left[W_{s}(x)\cap E_{i}\right]\supset W_{s}(\phi x)\cap E_{j}

Here, $W_{u}(x)$ and $W_{s}(x)$ are the unstable and stable manifolds of x, respectively. Here, $\operatorname {Int} E_{i}$ simply denotes the interior of $E_{i}$ . Letting $\partial$ denote the boundary operator, so that $\partial _{u}E_{i}$ is the boundary on the unstable side of $E_{i}$ (likewise for stable), then the last conditions may be written as: