Measure theoretic pressure and dimension formula for non-ergodic measures

Jialu Fang; Yongluo Cao; Yun Zhao

doi:10.3934/dcds.2020149

Article Contents

2020, Volume 40, Issue 5: 2767-2789. Doi: 10.3934/dcds.2020149

This issue Previous Article Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement Next Article Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions

Measure theoretic pressure and dimension formula for non-ergodic measures

1.
Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, China
2.
Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200062, China
3.
Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, Jiangsu, China

^* Corresponding author: Yun Zhao
^* Corresponding author: Yun Zhao

Received: May 31, 2019

Published: March 2020

The first and third author are partially supported by NSFC (11871361, 11790274). The second author is partially supported by NSFC (11790274, 11771317) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000)

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Abstract

This paper 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 Carathéodory-Pesin structure described in [13], and show that this quantity is equal to the essential supremum of the absolute value of free energy of the measures in an ergodic decomposition. Meanwhile, we define the measure theoretic pressure in another way by using separated sets, it is showed that this quantity is exactly the absolute value of 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 absolute value of free energy even if the measure is non-ergodic. As an application of the main results, 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.

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Mathematics Subject Classification: Primary: 37D35, 37C45; Secondary: 37A30.

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