The approximation of Lyapunov exponents by horseshoes for $C^1$-diffeomorphisms with dominated splitting

TL;DR

This paper extends Katok's Horseshoes construction for $C^1$-diffeomorphisms with dominated splitting, enabling approximation of Lyapunov exponents through horseshoes in systems with hyperbolic measures.

Contribution

It introduces an extension of Katok's Horseshoes construction for systems with dominated splitting, linking Oseledec subspaces to horseshoes.

Findings

Constructed horseshoes approximate Lyapunov exponents.

Established dominated splitting on horseshoes.

Linked Oseledec subspaces to horseshoe dynamics.

Abstract

Let $f$ be a $C^1$-diffeomorphism and $\mu$ be a hyperbolic ergodic $f$-invariant Borel probability measure with positive measure-theoretic entropy. Assume that the Oseledec splitting $$T_xM=E_1(x) \oplus\cdots\oplus E_s(x) \oplus E_{s+1}(x) \oplus\cdots\oplus E_l(x) $$ is dominated on the Oseledec basin $\Gamma$. We give extensions of Katok's Horseshoes construction. Moreover there is a dominated splitting corresponding to Oseledec subspace on horseshoes.