Open AccessExpansivity for measures on uniform spaces
Author(s) -
C. A. Morales,
Vı́ctor F. Sirvent
Publication year - 2015
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/tran/6555
Subject(s) - mathematics , measure (data warehouse) , expansive , separable space , σ finite measure , hausdorff measure , invariant measure , metric space , zero (linguistics) , outer measure , space (punctuation) , pure mathematics , probability measure , partition (number theory) , hausdorff distance , mathematical analysis , combinatorics , hausdorff dimension , ergodic theory , minkowski–bouligand dimension , fractal , computer science , composite material , linguistics , philosophy , materials science , compressive strength , fractal dimension , database
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.