Approximation Dynamics

TL;DR

This paper presents methods for approximating complex continuous dynamical systems on manifolds and Cantor sets using simpler, tractable systems with finite ergodic components, facilitating analysis and understanding.

Contribution

It introduces a novel approach to approximate continuous dynamical systems with finite chain components on p. l. manifolds and Cantor sets using simplicial and shift-like systems.

Findings

Effective approximation of continuous systems by tractable models

Finite ergodic decomposition for almost all points

Applicability to p. l. manifolds and Cantor sets

Abstract

We describe the approximation of a continuous dynamical system on a p. l. manifold or Cantor set by a tractable system. A system is tractable when it has a finite number of chain components and, with respect to a given full background measure, almost every point is generic for one of a finite number of ergodic invariant measures. The approximations use non-degenerate simplicial dynamical systems for p. l. manifolds and shift-like dynamical systems for Cantor Sets.