# Entropy of physical measures for $C^\infty$ smooth systems

## TL;DR

This paper establishes a relationship between the entropy of empirical measures and Lyapunov exponents for smooth dynamical systems, showing that typical points have empirical measures with entropy at least equal to the sum of positive Lyapunov exponents.

## Contribution

It proves that for smooth systems, almost every point's empirical measure has entropy at least the sum of positive Lyapunov exponents, linking entropy and Lyapunov exponents in a new way.

## Key findings

- Empirical measures at typical points have entropy ≥ sum of positive Lyapunov exponents.
- The result applies to $C^
abla$ smooth maps on compact manifolds.
- Provides a new lower bound for entropy in smooth dynamical systems.

## Abstract

For a $C^\infty$ map on a compact manifold we prove that for a Lebesgue randomly picked point x there is an empirical measure from $x$ with entropy larger than or equal to the sum of positive Lyapunov exponents at $x$.