Entropies of commuting transformations on Hilbert spaces

TL;DR

This paper extends ergodic theory to infinite-dimensional Hilbert spaces by establishing a Multiplicative Ergodic Theorem for commuting transformations, analyzing entropy, SRB measures, and applying results to specific group actions.

Contribution

It introduces a Multiplicative Ergodic Theorem for commutative transformations on infinite-dimensional Hilbert spaces and explores entropy and SRB measures in this context.

Findings

Established a Multiplicative Ergodic Theorem for commuting transformations.

Derived formulas for Pesin's entropy and Friedland's entropy in specific settings.

Analyzed SRB measures for finitely generated random transformations.

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.