# Measure theoretic pressure and dimension formula for non-ergodic measures

## TL;DR

This paper extends the concept of measure theoretic pressure to non-ergodic measures, establishing a new formula linking it to ergodic decompositions and applying it to determine Hausdorff dimensions in dynamical systems.

## Contribution

It introduces a novel measure theoretic pressure formula for non-ergodic measures and connects it to Hausdorff dimension calculations in dynamical systems.

## Key findings

- Measure theoretic pressure equals the essential supremum of free energy over ergodic components.
- The new pressure formula matches the free energy for ergodic measures.
- Hausdorff dimension of measures on conformal repellers is given by zeros of measure theoretic pressure.

## Abstract

This paper first studies the measure theoretic pressure of measures that are not necessarily ergodic. We define the measure theoretic pressure of an invariant measure (not necessarily ergodic) via the Carath\'{e}odory-Pesin structure described in \cite{Pes97}, and show that this quantity is equal to the essential supremum of the free energy of the measures in an ergodic decomposition. To the best of our knowledge, this formula is new even for entropy. Meanwhile, we define the measure theoretic pressure in another way by using separated sets, it is showed that this quantity is exactly the free energy if the measure is ergodic. Particularly, if the dynamical system satisfies the uniform separation condition and the ergodic measures are entropy dense, this quantity is still equal to the the free energy even if the measure is non-ergodic. As an application of the main result, we find that the Hausdorff dimension of an invariant measure supported on an average conformal repeller is given by the zero of the measure theoretic pressure of this measure. Furthermore, if a hyperbolic diffeomorphism is average conformal and volume-preserving, the Hausdorff dimension of any invariant measure on the hyperbolic set is equal to the sum of the zeros of measure theoretic pressure restricted to stable and unstable directions.