# A formula for the local metric pressure

## TL;DR

This paper introduces a new formula for local metric pressure, extending the Brin-Katok result for metric entropy, and demonstrates that non-atomic weak-Gibbs measures are equilibrium states.

## Contribution

It generalizes the concept of metric entropy to local metric pressure and provides a simple proof that certain measures are equilibrium states.

## Key findings

- The formula extends Brin-Katok's result to local metric pressure.
- Non-atomic weak-Gibbs measures are shown to be equilibrium states.
- Provides a straightforward proof for the equilibrium state property.

## Abstract

In this note we present a formula for the local metric pressure that generalizes Brin-Katok result for the metric entropy. As an application, we give a straightforward proof of the fact that non-atomic weak-Gibbs invariant probability measures are equilibrium states.