On bifurcation of statistical properties of partially hyperbolic endomorphisms

TL;DR

This paper presents an example of a smooth family of partially hyperbolic endomorphisms on the 2-torus where the SRB measure varies smoothly, but the sign of the central Lyapunov exponent changes, indicating a bifurcation in statistical properties.

Contribution

It provides a concrete example demonstrating bifurcation phenomena in the statistical properties of partially hyperbolic endomorphisms.

Findings

Existence of a smooth family of SRB measures on the 2-torus

Smooth variation of SRB measures with the system

Change in the sign of the central Lyapunov exponent

Abstract

We give an example of a path-wise connected open set of $C^\infty$ partially hyperbolic endomorphisms on the $2$-torus, on which the SRB measure exists for each system and varies smoothly depending on the system, while the sign of its central Lyapunov exponent does change.