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

TL;DR

This paper investigates how the entropy of a measurable map's invariant measure can be approximated from below using a converging sequence of invariant measures, highlighting the necessity of finite entropy assumptions.

Contribution

It provides a lower bound estimate for the Kolmogorov--Sinai entropy based on a sequence of invariant measures, emphasizing the importance of finite entropy partitions.

Findings

Lower bound estimate for entropy in terms of sequence measures

Finite entropy assumption is essential for the estimate

Explicit example demonstrating the necessity of the assumption

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. This result is obtained under the assumption that some generating partition has finite entropy. By an explicite example it is shown that, in general, this assumption cannot be removed.