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

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.