On r-Neutralized Entropy: Entropy Formula and Existence of Measures Attaining the Supremum

Abstract

In this article we study r-neutralized local entropy and derive some entropy formulas. For an ergodic hyperbolic measure of a smooth system, we show that the r-neutralized local entropy equals the Brin-Katok local entropy plus r times the pointwise dimension of the measure. We further establish the existence of ergodic measures that maximize the r-neutralized entropy for certain hyperbolic systems. Moreover, we construct a uniformly hyperbolic system, for which such measures are not unique. Finally, we present some rigidity results related to these ergodic measures.