Encodings of trajectories and invariant measures

TL;DR

This paper explores how symbolic representations of discrete dynamical systems on compact manifolds approximate actual trajectories and ergodic measures as the covering of the manifold becomes finer.

Contribution

It establishes the convergence of symbolic path sets to true trajectories and simple flows to ergodic measures as the covering diameter approaches zero.

Findings

Symbolic paths converge to system trajectories in Tychonoff topology.

Simple flows on cycles converge to ergodic measures in weak topology.

The results connect symbolic dynamics with invariant measures in a rigorous way.

Abstract

We consider a discrete dynamical system on a compact manifold M generated by a homeomorphism f. Let C = {M(i)} be a finite covering of M by closed cells. The symbolic image of a dynamical system is a directed graph G with vertices corresponding to cells in which vertices i and j are joined by an arc i to j if the image f(M(i)) intersects M(j). We show that the set of paths of the symbolic image converges to the set of trajectories of the system in the Tychonoff topology as the diameter of the covering tends to zero. For a cycle on G going through different vertices, a simple flow is by definition a uniform distribution on arcs of this cycle. We show that simple flows converge to ergodic measures in the weak topology as the diameter of the covering tends to zero.