Classification of one dimensional dynamical systems by countable structures

TL;DR

This paper investigates the complexity of classifying one-dimensional dynamical systems, showing that conjugacy relations can be characterized by countable structures, with implications for Hjorth's conjecture.

Contribution

It proves that conjugacy of interval dynamical systems is Borel bireducible to countable graph isomorphism, addressing a special case of Hjorth's conjecture.

Findings

Conjugacy of interval dynamical systems is Borel bireducible to countable graph isomorphism.

Conjugacy of Hilbert cube homeomorphisms is Borel bireducible to the universal orbit equivalence relation.

Addresses a specific case of Hjorth's conjecture on classifiability 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.