Symbolic extensions for 3-dimensional diffeomorphisms

TL;DR

This paper proves that all smooth three-dimensional diffeomorphisms have symbolic extensions, confirming a longstanding conjecture and advancing understanding of their dynamical complexity.

Contribution

It establishes that every $ ext{C}^r$ diffeomorphism on a 3D manifold admits a symbolic extension, solving a conjecture by Downarowicz and Newhouse.

Findings

Every $ ext{C}^r$ diffeomorphism on a 3D manifold admits a symbolic extension.

Answers positively a conjecture in dynamical systems.

Advances understanding of symbolic representations of complex systems.

Abstract

We prove that every $\mathcal{C}^{r}$ diffeomorphism with $r>1$ on a three-dimensional manifold admits symbolic extensions, i.e. topological extensions which are subshifts over a finite alphabet. This answers positively a conjecture of Downarowicz and Newhouse in dimension three.