Stable manifolds, Horseshoes and Lyapunov exponents for $C^1$ diffeomorphisms without domination

TL;DR

This paper extends nonuniform hyperbolic theory to certain $C^1$ diffeomorphisms lacking domination, establishing key dynamical structures like stable manifolds, horseshoes, and Lyapunov exponent approximation.

Contribution

It introduces a novel framework using resonance blocks to analyze hyperbolic behavior without domination in $C^1$ diffeomorphisms.

Findings

Existence of stable manifolds without domination

Construction of horseshoes in this setting

Approximation of Lyapunov exponents

Abstract

We develop the nonuniformly hyperbolic theory for $C^1$ diffeomorphisms admitting continuous invariant splitting without domination. This framework includes stable manifold theorems, shadowing and closing lemmas, the existence of horseshoes and the approximation of Lyapunov exponents. The foundation is a new family of resonance blocks, each arising as the forward limit set of a typical point at carefully chosen resonance times where expansion, contraction and a weak scale-dependent domination coexist.