Expansivity for measures on uniform spaces

Abstract

We define positively expansive and expansive measures on uniform spaces extending the analogous concepts on metric spaces. We show that such measures can exist for measurable or bimeasurable maps on compact non-Hausdorff uniform spaces. We prove that positively expansive probability measures on Lindelöf spaces are non-atomic and their corresponding maps eventually aperiodic. We prove that the stable classes of measurable maps have measure zero with respect to any positively expansive invariant measure. In addition, any measurable set where a measurable map in a Lindelöf uniform space is Lyapunov stable has measure zero with respect to any positively expansive inner regular measure. We conclude that the set of sinks of any bimeasurable map with canonical coordinates of a Lindelöf space has zero measure with respect to any positively expansive inner regular measure. Finally, we show that every measurable subset of points with converging semiorbits of a bimeasurable map on a separable uniform space has zero measure with respect to every expansive outer regular measure. These results generalize those found in works by Arbieto and Morales and by Reddy and Robertson.