Classification of one dimensional dynamical systems by countable structures

Abstract

We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of the Hjorth's conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube homeomorphisms is Borel bireducible to the universal orbit equivalence relation.

Analysis of Dynamical Behavior of One-Dimensional Real Maps: An Executable Dynamical Programming Software Approach

“…Furthermore, many studies [25] [26] [27] [28] investigated one-dimensional map characteristics under different conditions. Sushko et al discussed some basic concepts and definitions of non-smooth one-dimensional maps [29].…”