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Measure Theoretical Dynamical Systems
Measures on Compact Metric Spaces
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assume atoms Bernoulli shift Borel measure Borel partition Borel set called canonical coordinates choose Clearly construction converges convex Corollary countable defined Definition denote dense diam diffeomorphism disjoint easy eigenvalue ergodic measures ergodic theory exists f.t. subshift finite type follows h-expansive hence homeomorphism htop(T implies integers intrinsically ergodic invariant measures Lebesgue spaces lemma Let X,T lim sup M-generator m.f.t.-subshift m.t. dynamical system Markov partition matrix maximal entropy measure theoretic metric space minimal N-blocks n,e)-separated nonempty nonwandering nonwandering set o-algebra obtains obvious occurs open cover open sets periodic points Proof Proposition resp Rohlin set satisfies the specification sequence set F specification property strictly ergodic strongly mixing subset subshift of finite subshift of order T-invariant theorem topo topological dynamical system topological entropy topologically mixing topologically transitive transformation uniquely ergodic v-continuity weakly mixing write