A local variational principle for random bundle transformations

TL;DR

This paper develops a local variational principle for random bundle transformations by introducing new local entropy concepts and establishing inequalities and relations among them.

Contribution

It introduces local topological and measure-theoretic entropy for random bundle transformations and proves a local variational principle relating these entropies.

Findings

Established a variational inequality for random local entropy.

Proved a local variational principle in random dynamical systems.

Defined new notions of local entropy for random transformations.

Abstract

We introduce local topological entropy $h_{\text{top}}(T, \mathcal{U})$ and two kinds of local measure-theoretic entropy $h_{\mu}^{(r)-}(T,\mathcal{U})$ and $h_{\mu}^{(r)+}(T,\mathcal{U})$ for random bundle transformations. We derive a variational inequality of random local entropy for $h_{\mu}^{(r)+}(T,\mathcal{U})$. As an application of such relation we prove a local variational principle in random dynamical system.