Expansividad para medidas en espacios uniformes
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Autores
Salcedo Sora, Juan Carlos
Director
Bautista Díaz, Serafín
Tipo de contenido
Trabajo de grado - Maestría
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EspañolFecha de publicación
2014-12-05
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Con el objetivo principal de realizar una reconstrucci´on detallada del art´ıculo [14] intentando responder los diversos interrogantes que surgen en el mismo, definimos medidas expansivas y positivamente expansivas sobre espacios uniformes, extendiendo el concepto an´alogo sobre espacios m´etricos desarrollados en [1] y [15]. Mostramos que tales medidas pueden existir para funciones medibles o bimedibles sobre espacios uniformes compactos no Hausdorff, demostramos que las medidas de probabilidad positivamente expansivas sobre espacios de Lindel¨of son no at´omicas y sus correspondientes funciones eventualmente aperi´odicas (y por tanto aperi´odicas), mostramos que la clase estable de funciones medibles tienen medida cero con respecto a cualquier medida invariante positivamente expansiva. Adicionalmente, cualquier conjunto medible en donde una funci´on medible en un espacio de Lindel¨of es Lyapunov estable tiene medida cero con respecto a cualquier medida regular interior positivamente expansiva. Concluimos que el conjunto de sumideros de cualquier funci´on bimedible con coordenadas can´onicas de un espacio Lindel¨of tiene medida cero con respecto a cualquier medida regular interior positivamente expansiva. Finalmente, mostramos que todo subconjunto medible de puntos con semi´orbitas convergentes de una funci´on bimedible sobre un espacio uniforme separable tiene medida cero con respecto a toda medida regular exterior expansiva, lo cual generaliza los resultados establecidos en [18] y [1].
Abstract: With the main goal of a detailed reconstruction of the article [14], we try to answer the questions that arise in it, we define positively expansive and expansive measures on uniform spaces extending the analogous concept on metric spaces analogue developed in [1] and [15]. We show that such measures can exist for measurable or bimeasurable functions on compact non-Hausdorff uniform spaces, we show that positively expansive probability measures on Lindel¨of spaces are non-atomic and their corresponding maps eventually aperiodic (and hence aperiodic), we show that the stable classes of measurable maps have measure zero with respect to any positively expansive invariant measure. Additionally, any measurable set where a measurable map in a Lindel¨of 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 Lindel¨of 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, which generalizes the results established in [18] and [1].
Abstract: With the main goal of a detailed reconstruction of the article [14], we try to answer the questions that arise in it, we define positively expansive and expansive measures on uniform spaces extending the analogous concept on metric spaces analogue developed in [1] and [15]. We show that such measures can exist for measurable or bimeasurable functions on compact non-Hausdorff uniform spaces, we show that positively expansive probability measures on Lindel¨of spaces are non-atomic and their corresponding maps eventually aperiodic (and hence aperiodic), we show that the stable classes of measurable maps have measure zero with respect to any positively expansive invariant measure. Additionally, any measurable set where a measurable map in a Lindel¨of 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 Lindel¨of 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, which generalizes the results established in [18] and [1].