Weak approximation of an invariant measure and a low boundary of the entropy

TL;DR

This paper proposes a method to estimate the entropy of an invariant measure for a measurable map by analyzing the convergence of a sequence of invariant measures and their entropies.

Contribution

It introduces a lower bound estimate for the Kolmogorov--Sinai entropy based on the convergence of invariant measures and their entropies.

Findings

Provides a lower estimate for entropy in terms of converging measures

Establishes a connection between measure convergence and entropy approximation

Enhances understanding of entropy behavior under measure limits

Abstract

For a measurable map $T$ and a sequence of $T$-invariant probability measures $\mu_n$ that converges in some sense to a $T$-invariant probability measure $\mu$, an estimate from below for the Kolmogorov--Sinai entropy of $T$ with respect to $\mu$ is suggested in terms of the entropies of $T$ with respect to $\mu_1$, $\mu_2$, \dots.