Fiber entropy and conditional variational principles in compact non-metrizable spaces

Tomasz Downarowicz ¹   ; Jacek Serafin ¹

¹ Institute of Mathematics Wrocław Technical University Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland

Fundamenta Mathematicae, Tome 172 (2002) no. 3, pp. 217-247

Résumé

We consider a pair of topological dynamical systems on compact Hausdorff (not necessarily metrizable) spaces, one being a factor of the other. Measure-theoretic and topological notions of fiber entropy and conditional entropy are defined and studied. Abramov and Rokhlin's definition of fiber entropy is extended, using disintegration. We prove three variational principles of conditional nature, partly generalizing some results known before in metric spaces: (1) the topological conditional entropy equals the supremum of the topological fiber entropy over the factor, which also equals the supremum of the topological fiber entropy given a measure over all invariant measures on the factor, (2) the topological fiber entropy given a measure equals the supremum of the measure-theoretic conditional entropy over all invariant measures on the larger system projecting to the given one. Combining the above, we get (3) the topological conditional entropy equals the supremum of the measure-theoretic conditional entropy over all invariant measures. A tail entropy of a measure is introduced in totally disconnected spaces. As an application of our variational principles it is proved that the tail entropy estimates from below the “defect of upper semicontinuity” of the entropy function.

Détail

DOI : 10.4064/fm172-3-2

Keywords: consider pair topological dynamical systems compact hausdorff necessarily metrizable spaces being factor other measure theoretic topological notions fiber entropy conditional entropy defined studied abramov rokhlins definition fiber entropy extended using disintegration prove three variational principles conditional nature partly generalizing results known before metric spaces topological conditional entropy equals supremum topological fiber entropy factor which equals supremum topological fiber entropy given measure invariant measures factor topological fiber entropy given measure equals supremum measure theoretic conditional entropy invariant measures larger system projecting given combining above get topological conditional entropy equals supremum measure theoretic conditional entropy invariant measures tail entropy measure introduced totally disconnected spaces application variational principles proved tail entropy estimates below defect upper semicontinuity entropy function