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Título: =Continuity of conditional measures associated to measure-preserving semiflows
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Publicación seriada
Referencias AnalíticasReferencias Analíticas
Autor: M. McClendon, David
Título: Continuity of conditional measures associated to measure-preserving semiflows
Páginas/Colación: pp. 331-341
Fecha: January 2009
Transactions of the American Mathematical Society Vol. 361, no. 1 January 2009
Información de existenciaInformación de existencia

Palabras Claves: Palabras: MEASURE-PRESERVING SEMIFLOWS MEASURE-PRESERVING SEMIFLOWS

Resumen
It is proved herein that any absolute minimizer for a suitable Hamiltonian is a viscosity solution of the Aronsson equation:

Let $ X$be a standard probability space and $ T_t$a measure-preserving semiflow on $ X$. We show that there exists a set $ X_0$of full measure in $ X$such that for any $ x \in X_0$and $ t \geq 0$there are measures $ \mu_{x,t}^+$and $ \mu_{x,t}^-$which for all but a countable number of $ t$give a distribution on the set of points $ y$such that $ T_t(y) = T_t(x)$. These measures arise by taking weak$ ^*-$limits of suitable conditional expectations. Say that a point $ x$has a measurable orbit discontinuity at time $ t_0$if either $ \mu_{x,t}^+$or $ \mu_{x,t}^-$is weak$ ^*-$discontinuous in $ t$at $ t_0$. We show that there exists an invariant set of full measure in $ X$such that any point in this set has at most countably many measurable orbit discontinuities. Furthermore we show that if $ x$has a measurable orbit discontinuity at time 0, then $ x$has an orbit discontinuity at time 0 in the sense of Orbit discontinuities and topological models for Bordel

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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