# Topological pressure for conservative $C^1$-diffeomorphisms with no dominated splitting

## TL;DR

This paper establishes formulas for topological pressure in generic conservative $C^1$-diffeomorphisms without dominated splitting, showing the absence of positive entropy equilibrium states and identifying phase transition phenomena.

## Contribution

It provides new formulas for topological pressure, proves continuity results, and describes phase transitions for generic conservative diffeomorphisms without dominated splitting.

## Key findings

- No equilibrium states with positive entropy for generic cases.
- Existence of zero-entropy equilibrium states for certain potentials.
- Identification of a phase transition point in the family of potentials.

## Abstract

We prove three formulas for computing topological pressure of $C^1$-generic conservative diffeomorphism and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that there is no equilibrium states with positive measure theoretic entropy. In particular, for hyperbolic potentials, there is no equilibrium states. For $C^1$ generic conservative diffeomorphism on compact surfaces with no dominated splitting and $\phi_m(x):=-\frac{1}{m}\log \Vert D_x f^m\Vert, m \in \mathbb{N}$, we show that there exist equilibrium states with zero entropy and there exists a transition point $t_0$ for the family $\lbrace t \phi_m(x)\rbrace_{t\geq 0}$, such that there is no equilibrium states for $ t \in [0, t_0)$ and there is an equilibrium state for $t \in [t_0,+\infty)$.