Extremal exponents of random products of conservative diffeomorphisms

TL;DR

This paper demonstrates that in a broad class of conservative dynamical systems, the extremal Lyapunov exponents are typically non-zero, indicating prevalent non-uniform hyperbolicity in these systems.

Contribution

It establishes that non-vanishing extremal Lyapunov exponents are generic in a $C^1$-open and $C^{r}$-dense subset of ergodic systems of conservative diffeomorphisms.

Findings

Non-zero extremal Lyapunov exponents are generic in the considered class.

Non-uniform hyperbolic systems form a $C^1$-open and $C^{r}$-dense subset.

Most ergodic random products of conservative surface diffeomorphisms exhibit non-vanishing exponents.

Abstract

We show that for a $C^1$-open and $C^{r}$-dense subset of the set of ergodic iterated function systems of conservative diffeomorphisms of a finite-volume manifold of dimension $d\geq 2$, the extremal Lyapunov exponents do not vanish. In particular, the set of non-uniform hyperbolic systems contains a $C^1$-open and $C^r$-dense subset of ergodic random products of i.i.d. conservative surface diffeomorphisms.