Recurrent Points and Transition Functions Acting on Continuous Functions.
CALIFORNIA UNIV SAN DIEGO LA JOLLA
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Omega is assumed to be a locally compact separable metric space with P a transition function taking bounded continuous functions on omega into continuous functions. A point x is said to be a point of sure return if for each neighborhood M of x PX sub k belongs to M for some k or 1 X sub 0 1, and a recurrent point if for each neighborhood M of x PX sub k belongs to M for infinitely many j or 1 X sub 0 x 1, where X sub k is a Markov process with transition function P. It is shown that if x is a point of sure return, it is recurrent. Various implications of recurrence for points are investigated. Corresponding notions of conservative and dissipative points based in part on topological ideas are considered and compared with related concepts introduced by Foguel. Author
- Statistics and Probability