A measure-theoretic expansion exponent
C. A. Morales
TL;DR
The paper defines a measure-theoretic expansion exponent $E_\mu(T)$ for Borel probability measures and proves the map exponent $E(T)$ equals the minimum of the measure exponents: $E(T)=\min_\mu E_\mu(T)$. This yields that small-distance expansion holds exactly when all measures have positive exponent, and that nonatomic invariant measures with positive exponent are positively expansive; for ergodic invariant measures on compact spaces, the Kolmogorov-Sinai entropy satisfies $h_\mu(T)\ge \overline{dim}_B(\mu)\,E_\mu(T)$, linking expansion to information-theoretic complexity. The results connect measure-theoretic concentration with classical expansion notions, and imply positive entropy under appropriate capacity and expansion conditions via Katok’s entropy framework. These findings provide a unified, non-differentiable framework for expansion and its entropy consequences.
Abstract
The expansion exponent (or expansion constant) for maps was introduced by Schreiber in \cite{s}. In this paper, we introduce the analogous exponent for measures. We shall prove the following results: The expansion exponent of a measurable maps is equal to the minimum of the expansion exponent taken over the Borel probability measures. In particular, a map expands small distances (in the sense of Reddy \cite{r}) if and only if every Borel probability has positive expansion exponent. Any nonatomic invariant measure with positive expansion exponent is positively expansive in the sense of \cite{m}. For ergodic invariant measures, the Kolmogorov-Sinai entropy is bounded below by the product of the expansion exponent and the measure upper capacity. As a consequence, any ergodic invariant measure with both positive upper capacity and positive expansion exponent must have positive entropy.