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Abstract

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.