Rigidity of center Lyapunov exponents for Anosov diffeomorphisms on 3-torus

TL;DR

This paper proves that conjugate Anosov diffeomorphisms on the 3-torus with foliation-preserving conjugacy have identical center Lyapunov exponents at periodic points, extending previous results to partially hyperbolic cases.

Contribution

It establishes the rigidity of center Lyapunov exponents under conjugacy that preserves strong stable foliations for Anosov and partially hyperbolic diffeomorphisms on T3.

Findings

Center Lyapunov exponents coincide for conjugate Anosov diffeomorphisms.

Results extend to partially hyperbolic diffeomorphisms derived from Anosov.

Conjugacy preserving strong stable foliation implies rigidity of exponents.

Abstract

Let f and g be two Anosov diffeomorphisms on T3 with three-subbundles partially hyperbolic splittings where the weak stable subbundles are considered as center subbundles. Assume that f is conjugate to g and the conjugacy preserves the strong stable foliation, then their center Lyapunov exponents of corresponding periodic points coincide. This is the converse of the main result of Gogolev and Guysinsky in [9]. Moreover, we get the same result for partially hyperbolic diffeomorphisms derived from Anosov on T3.