Rigidity of Lyapunov exponents for derived from Anosov diffeomorphisms

TL;DR

This paper establishes bounds on the sums of Lyapunov exponents for certain volume-preserving, partially hyperbolic diffeomorphisms homotopic to linear Anosov automorphisms, extending rigidity results to derived from Anosov and non-uniformly hyperbolic systems.

Contribution

It proves bounds on Lyapunov exponent sums for derived from Anosov systems, advancing the understanding of their rigidity properties.

Findings

Sum of positive Lyapunov exponents is bounded above by that of the linearization.

Sum of negative Lyapunov exponents is bounded below by that of the linearization.

Results apply to classes of non-uniformly hyperbolic systems with dominated splitting.

Abstract

For a class of volume preserving partially hyperbolic diffeomorphisms (or non-uniformly Anosov) $f\colon {\T}^d\rightarrow{\T}^d$ homotopic to linear Anosov automorphism, we show that the sum of the positive (negative) Lyapunov exponents of $f$ is bounded above (resp. below) by the sum of the positive (resp. negative) Lyapunov exponents of its linearization. We show this for some classes of derived from Anosov (DA) and non-uniformly hyperbolic systems with dominated splitting, in particular for examples described by C. Bonatti and Viana. The results in this paper address a flexibility program by J. Bochi, A. Katok and F. Rodriguez Hertz.