# Unstable Metric Pressure of Partially Hyperbolic Diffeomorphisms with sub-additive Potentials

## TL;DR

This paper introduces and analyzes unstable measure theoretic and topological pressures for partially hyperbolic diffeomorphisms with sub-additive potentials, establishing their equivalence and relation to entropy and Lyapunov exponents.

## Contribution

It defines unstable pressures for partially hyperbolic systems with sub-additive potentials and proves their equivalence across different definitions and their relation to entropy and Lyapunov exponents.

## Key findings

- Unstable metric pressure equals unstable measure theoretic entropy plus Lyapunov exponents.
- All definitions of unstable metric pressure coincide for ergodic measures.
- The framework extends pressure concepts to partially hyperbolic systems with sub-additive potentials.

## Abstract

In this paper, we define and study unstable measure theoretic pressure for C^1-smooth partially hyperbolic diffeomorphisms with sub-additive potentials. For any ergodic measure, we show that this unstable metric pressure equals the corresponding unstable measure theoretic entropy plus the Lyapunov exponents with respect to the ergodic measure. On the other hand, we also define unstable topological and metric pressure in terms of the Bowen's picture and the capacity picture. We show that all these definitions of unstable metric pressure actually coincide for any ergodic measure.