Entropies of commuting transformations on Hilbert spaces

Zhiming Li; Yujun Zhu

doi:10.3934/dcds.2020246

Article Contents

2020, Volume 40, Issue 10: 5795-5814. Doi: 10.3934/dcds.2020246

This issue Previous Article On the effects of firing memory in the dynamics of conjunctive networks Next Article Evolution of dispersal in advective homogeneous environments

Entropies of commuting transformations on Hilbert spaces

Zhiming Li^1, , and
Yujun Zhu^2,

1.
School of Mathematics, Northwest University, Xi'an 710127, China
2.
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

^* Corresponding author: Zhiming Li
^* Corresponding author: Zhiming Li

Received: September 2019

Revised: May 2020

Published: June 2020

The first author is supported by NSFC (No: 11871394) and Natural Science Foundation of Shaanxi Province (2020JC-39), the second author is supported by NSFC (No: 11771118)

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Abstract

By establishing Multiplicative Ergodic Theorem for commutative transformations on a separable infinite dimensional Hilbert space, in this paper, we investigate Pesin's entropy formula and SRB measures of a finitely generated random transformations on such space via its commuting generators. Moreover, as an application, we give a formula of Friedland's entropy for certain $ C^{2} $ $ \mathbb{N}^2 $-actions.

Keywords:

SRB measures,
metric entropy,
Pesin's entropy formula,
$ \mathbb{N}^2 $-action,
infinite dimensional random dynamical system.

Mathematics Subject Classification: Primary:37H15, 37A35, 37C85;Secondary:37H99, 37D20.

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