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https://hdl.handle.net/2142/86781
Description
Title
Convergence in Ergodic Theory
Author(s)
Argiris, Georgios
Issue Date
2001
Doctoral Committee Chair(s)
Rosenblatt, Joseph
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
We investigate two problems involving convergence in ergodic theory. The first problem is the following: Given a measure preserving transformation T and a weight function w(alpha) → 0 as alpha → 0, is there a p > 0 such that the expression w(alpha)#{ n : 1n k=1nfT kx >a } have a limit, a.s. or in norm, as alpha → 0 for all functions f ∈ Lp0 [0,1]? No, we show. Here # denotes counting measure and f's are taken to be mean-zero functions. We also consider similar questions for the more general operator w(alpha)#{n : 1nq k=1n f(Tk(x)) > alpha}, q > 1. The second problem addressed is to give arithmetic and probabilistic characterizations on the integer sequence ( nk) such that the series of ergodic differences k=1infinity ( Ank+1f-Ankf ), where An denotes the usual ergodic averages, converges unconditionally for all functions f in some Lp space.
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