Lectures on Ergodic TheoryLectures for a graduate course in ergodic theory at the University of Minnesota, fall and winter 1971/72 quarters. |
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2-shift A₁ assume automorphism B₁ B₂ bisequence Borel sets closed invariant compact metric compact T2 completes the proof consider constant a.e. contradiction converges Corollary countable D₁ D₂ defined Definition diam disjoint eigenfunction eigenvalue equicontinuous ergodic iff ergodic measurable Ergodic Theory Exercise exists expansive finite partition flow X,T h T,e h(el Haar measure Hence homeomorphism implies Individual Ergodic Theorem infinitely integer isomorphic L₂ Lebesgue measure Lemma Lemma 94 Let f lim h limsup log₂ measure preserving measure-theoretic minimal cardinality minimal subcover need only show Note open cover P₁ P₂ probability space prove result satisfies sequence spanning set spatially strictly ergodic strongly mixing subflow sup h Suppose X,T T-invariant Theorem 117 Theorem 76 Theorem 83 topological entropy weakly mixing X₁