Formula of Entropy along Unstable Foliations for $C^1$ Diffeomorphisms with Dominated Splitting

TL;DR

This paper investigates the metric entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting, establishing analogues of Ruelle's inequality and Pesin's formula for these hierarchies.

Contribution

It introduces new formulas relating metric entropy and Lyapunov exponents along unstable foliations in systems with dominated splitting.

Findings

Established analogues of Ruelle's inequality.

Derived Pesin's formula for unstable foliations.

Enhanced understanding of entropy in $C^1$ dynamical systems.

Abstract

Metric entropies along a hierarchy of unstable foliations are investigated for $C^1$ diffeomorphisms with dominated splitting. The analogues of Ruelle's inequality and Pesin's formula, which relate the metric entropy and Lyapunov exponents in each hierarchy, are given.