# Ergodic properties of partially hyperbolic diffeomorphisms with topological neutral center

## TL;DR

This paper investigates the ergodic and metric properties of certain partially hyperbolic diffeomorphisms with a neutral center, revealing conditions under which the system exhibits Bernoulli behavior and invariance principles.

## Contribution

It introduces a trichotomy for center conditional measures and establishes new regularity and ergodic properties for systems with neutral center foliation.

## Key findings

- Center foliation is leafwise absolutely continuous under full support conditionals.
- The system is Bernoulli in the $C^{1+}$ case when conditionals have full support.
- An invariance principle is established, linking regularity of the center foliation to accessibility hypotheses.

## Abstract

In this work we obtain some metric and ergodic properties of $C^{1+}$ partially hyperbolic diffeomorphisms with one-dimensional topological neutral center, mainly regarding the behavior of its center foliation. Based on a trichotomy for the center conditional measures of any invariant ergodic measure, we show that if these conditionals have full support, then the center foliation is leafwise absolutely continuous, the diffeomorphism is Bernoulli in the $C^{1+}$ case, and an invariance principle occurs in the sense that M may be covered by a finite number of open sets where the system of center conditionals is continuous and su-invariant. Using this invariance principle we show that if a local accessibility hypothesis occurs then the center foliation must be as regular as the partially hyperbolic dynamics.