Central Lyapunov exponent of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$

TL;DR

This paper constructs volume-preserving partially hyperbolic diffeomorphisms on the 3-torus with atypical central Lyapunov exponents, including examples with opposite behavior to their linearization and zero exponents almost everywhere.

Contribution

It introduces new examples of partially hyperbolic diffeomorphisms on $ oro{3}$ with unusual central Lyapunov exponent behavior, challenging existing expectations.

Findings

Existence of volume-preserving partially hyperbolic diffeomorphisms with opposite central Lyapunov behavior to linearization.

Construction of examples with zero central Lyapunov exponent almost everywhere.

Examples are isotopic to Anosov diffeomorphisms with non-compact central leaves.

Abstract

In this paper we construct some "pathological" volume preserving partially hyperbolic diffeomorphisms on $\toro{3}$ such that their behaviour in small scales in the central direction (Lyapunov exponent) is opposite to the behavior of their linearization. These examples are isotopic to Anosov. We also get partially hyperbolic diffeomorphisms isotopic to Anosov (consequently with non-compact central leaves) with zero central Lyapunov exponent at almost every point.