Symbolic images and invariant measures of dynamical systems

Abstract

AbstractLet f be a homeomorphism of a compact manifold M. The Krylov–Bogoloubov theorem guarantees the existence of a measure that is invariant with respect to f. The set of all invariant measures ℳ(f) is convex and compact in the weak topology. The goal of this paper is to construct the set ℳ(f). To obtain an approximation of ℳ(f), we use the symbolic image with respect to a partition C={M(1),M(2),…,M(n)} of M. A symbolic image G is a directed graph such that a vertex i corresponds to the cell M(i) and an edge i→j exists if and only if f(M(i))∩M(j)≠0̸. This approach lets us apply the coding of orbits and symbolic dynamics to arbitrary dynamical systems. A flow on the symbolic image is a probability distribution on the edges which satisfies Kirchhoff’s law at each vertex, i.e. the incoming flow equals the outgoing one. Such a distribution is an approximation to some invariant measure. The set of flows on the symbolic image G forms a convex polyhedron ℳ(G) which is an approximation to the set of invariant measures ℳ(f). By considering a sequence of subdivisions of the partitions, one gets sequence of symbolic images Gk and corresponding approximations ℳ(Gk) which tend to ℳ(f) as the diameter of the cells goes to zero. If the flows mk on each Gk are chosen in a special manner, then the sequence {mk} converges to some invariant measure. Every invariant measure can be obtained by this method. Applications and numerical examples are given.