Neutralized local entropy, and dimension bounds for invariant measurs

TL;DR

This paper introduces a new notion of local entropy called neutralized local entropy, demonstrating its equivalence to Brin-Katok entropy and applying it to establish lower bounds on the point-wise dimension of invariant measures.

Contribution

It defines neutralized local entropy, compares it with existing entropy notions, and uses it to derive lower bounds on invariant measure dimensions.

Findings

Neutralized local entropy coincides with Brin-Katok local entropy almost everywhere.

The new entropy measure can be computed via simple geometric descriptions.

Application to lower point-wise dimension bounds for invariant measures.

Abstract

We introduce a notion of a point-wise entropy of measures (i.e local entropy) called neutralized local entropy, and compare it with the Brin-Katok local entropy. We show that the neutralized local entropy coincides with Brin-Katok local entropy almost everywhere. Neutralized local entropy is computed by measuring open sets with a relatively simple geometric description. Our proof uses a measure density lemma for Bowen balls, and a version of a Besicovitch covering lemma for Bowen balls. As an application, we prove a lower point-wise dimension bound for invariant measures, complementing the previously established bounds for upper point-wise dimension.