Abstract
AbstractWe discuss in this chapter several additional topics of ergodic theory. In particular, we establish the existence of finite invariant measures for any continuous transformation of a compact metric space. We also discuss the notions of unique ergodicity and mixing. In particular, we show how unique ergodicity can be characterized in terms of uniform convergence in Birkhoff’s ergodic theorem. We conclude this chapter with brief introductions to symbolic dynamics and topological dynamics, with emphasis on their relations to ergodic theory. This includes the construction of Markov measures and Bernoulli measures, as well as the relation between Markov measures and topological Markov chains. We also show that each ergodic property implies a corresponding topological property.KeywordsInvariant MeasureErgodic TheoryStochastic MatrixSymbolic DynamicErgodic PropertyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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