A C1 generic condition for existence of symbolic extensions of volume preserving diffeomorphisms

TL;DR

This paper establishes that for volume-preserving diffeomorphisms, the existence of symbolic extensions is equivalent to partial hyperbolicity, confirming Bonatti's conjecture in this setting.

Contribution

It proves a C1-generic condition linking symbolic extensions to partial hyperbolicity for volume-preserving diffeomorphisms, confirming Bonatti's conjecture.

Findings

Symbolic extension exists iff the diffeomorphism is partial hyperbolic.

In the non-Anosov case, robust heterodimensional cycles are dense.

The result applies to C1-generic volume-preserving diffeomorphisms.

Abstract

We prove that a C1-generic volume preserving diffeomorphism has a symbolic extension if and only if this diffeomorphism is partial hyperbolic. This result is obtained by means of good dichotomies. In particular, we prove Bonatti's conjecture in the volume preserving scenario. More precisely, in the complement of Anosov diffeomorphisms we have densely robust heterodimensional cycles.