Robust vanishing of all Lyapunov exponents for iterated function systems

TL;DR

This paper constructs open sets of iterated function systems on manifolds that support ergodic measures with all zero Lyapunov exponents, including measures approximated by periodic orbits and measures with positive entropy.

Contribution

It introduces new methods to produce non-hyperbolic ergodic measures with zero Lyapunov exponents in IFSs on manifolds, expanding understanding of non-hyperbolic dynamics.

Findings

Existence of $C^2$-open sets with ergodic measures of zero exponents

Construction of measures approximated by periodic orbits

Presence of $C^1$-open sets with positive entropy measures of zero exponents

Abstract

Given any compact connected manifold $M$, we describe $C^2$-open sets of iterated functions systems (IFS's) admitting fully-supported ergodic measures whose Lyapunov exponents along $M$ are all zero. Moreover, these measures are approximated by measures supported on periodic orbits. We also describe $C^1$-open sets of IFS's admitting ergodic measures of positive entropy whose Lyapunov exponents along $M$ are all zero. The proofs involve the construction of non-hyperbolic measures for the induced IFS's on the flag manifold.