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DYNAMICAL SYSTEMS AND AMENABLE GROUPS
ENTROPY OF ABSTRACT DYNAMICAL SYSTEMS
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a-algebra abstract dynamical system action of G ameaning filter amenable group atoms automorphisms averages Bernoulli schemes compact Hausdorff space compact set conditional expectations Consider contained continuous function convergence convex and compact convex set Corollary countable group decomposition defined Definition denote dynamical system element g exists extremal points fixed-point property following equality Fourier transform function f function on F(G Gibbs measures given greatest lower bound group G Hausdorff space homeomorphisms integral invariant capacity invariant element invariant Radon probability isomorphic least upper bound limit linear functional Math mean entropy non-negative norm one-to-one mappings open cover open sets phase transition positive real number probability space probability vector Proof Proposition pseudo-norm quasi-invariant Radon probability measure ratio real function real vector space Remark result right translations sequence strictly positive strongly subadditive sublinear function subset subspace Sup y(f topological dynamical system topological entropy total orders upper semi-continuous function variational principle verifies