Dimension approximation in smooth dynamical systems

TL;DR

This paper demonstrates how the Lyapunov and Hausdorff dimensions of certain smooth dynamical systems can be approximated by the dimensions of horseshoes, providing a bridge between complex measures and simpler hyperbolic sets.

Contribution

It introduces methods to approximate Lyapunov and Hausdorff dimensions of ergodic measures in smooth dynamical systems using sequences of horseshoes.

Findings

Lyapunov dimension can be approximated by Carathéodory singular dimension of horseshoes.

Hausdorff and box dimensions of measures can be approximated by those of horseshoes.

Results hold for systems with dominated Oseledet's splitting.

Abstract

For a non-conformal repeller $\Lambda$ of a $C^{1+\alpha}$ map $f$ preserving an ergodic measure $\mu$ of positive entropy, this paper shows that the Lyapunov dimension of $\mu$ can be approximated gradually by the Carath\'{e}odory singular dimension of a sequence of horseshoes. For a $C^{1+\alpha}$ diffeomorphism $f$ preserving a hyperbolic ergodic measure $\mu$ of positive entropy, if $(f, \mu)$ has only two Lyapunov exponents $\lambda_u(\mu)>0>\lambda_s(\mu)$, then the Hausdorff or lower box or upper box dimension of $\mu$ can be approximated by the corresponding dimension of the horseshoes $\{\Lambda_n\}$. The same statement holds true if $f$ is a $C^1$ diffeomorphism with a dominated Oseledet's splitting with respect to $\mu$.