Non-existence of Lyapunov exponents in the Newhouse domain

TL;DR

This paper demonstrates that in the Newhouse domain of surface diffeomorphisms, non-existence of Lyapunov exponents occurs densely and persistently due to robust homoclinic tangencies, using advanced dynamical systems techniques.

Contribution

It establishes the persistent non-existence of Lyapunov exponents in the Newhouse domain, extending understanding of irregular behavior in dynamical systems.

Findings

Non-existence of Lyapunov exponents is dense in the Newhouse domain.

This phenomenon persists under robust homoclinic tangencies.

Constructs specific diffeomorphisms with oscillatory return times near tangencies.

Abstract

We show that within the Newhouse domain of $C^r$ surface diffeomorphisms ($r \in [2,\infty )$), there exists a dense subset $\mathcal D$ such that for any $f \in \mathcal D$, Lyapunov exponents fail to exist for all points in some open set $U$ and all nonzero tangent vectors in some open cone $V \subset \mathbb{R}^2$. This demonstrates that the non-existence of Lyapunov exponents is a persistent phenomenon in the setting of robust homoclinic tangencies. The proof relies on constructing diffeomorphisms exhibiting specific oscillatory return times near a homoclinic tangency, incorporating techniques from Newhouse theory and recent results on Lyapunov irregularity, alongside several refinements and new arguments.