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From discrete- to continuous-time ergodic theorems

Published online by Cambridge University Press:  16 December 2011

V. BERGELSON ,
A. LEIBMAN  and
C. G. MOREIRA
V. BERGELSON
Affiliation:
Department of Mathematics, The Ohio State University, OH 43221, USA (email: vitaly@math.ohio-state.edu, leibman@math.ohio-state.edu)
A. LEIBMAN
Affiliation:
Department of Mathematics, The Ohio State University, OH 43221, USA (email: vitaly@math.ohio-state.edu, leibman@math.ohio-state.edu)
C. G. MOREIRA
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brasil (email: gugu@impa.br)
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Abstract

We introduce methods that allow us to derive continuous-time versions of various discrete-time ergodic theorems. We then illustrate these methods by giving simple proofs and refinements of some known results as well as establishing new results of interest.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 383 - 426
Copyright
Copyright © Cambridge University Press 2011

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References

[A]Assani, I.. Multiple recurrence and almost sure convergence for weakly mixing dynamical systems. Israel J. Math. 163 (1998), 111–124.CrossRefGoogle Scholar
[Au1]Austin, T.. On the norm convergence of nonconventional ergodic averages. Ergod. Th. & Dynam. Sys. 30(2) (2010), 321–338.CrossRefGoogle Scholar
[Au2]Austin, T.. Norm convergence of continuous-time polynomial multiple ergodic averages. Ergod. Th. & Dynam. Sys. to appear.Google Scholar
[BK]Bergelson, V. and Håland, I.. Knutson weak mixing implies weak mixing of higher orders along tempered functions. Ergod. Th. & Dynam. Sys. 29 (2009), 1375–1416.CrossRefGoogle Scholar
[BL]Bergelson, V. and Leibman, A.. Distribution of values of bounded generalized polynomials. Acta Math. 198 (2007), 155–230.CrossRefGoogle Scholar
[BLM]Bergelson, V., Leibman, A. and McCutcheon, R.. Polynomial Szemeredi theorem for countable modules over integral domains and finite fields. J. Anal. Math. 95 (2005), 243–296.CrossRefGoogle Scholar
[BM]Bergelson, V. and McCutcheon, R.. An ergodic IP polynomial Szemeredi theorem. Mem. Amer. Math. Soc. 146 (2000), viii+106pp.Google Scholar
[BiKo]Birkhoff, G. D. and Koopman, B. O.. Recent contributions to the ergodic theory. Proc. Natl. Acad. Sci. USA 18 (1932), 279–282.CrossRefGoogle Scholar
[Bo1]Bourgain, J.. Double recurrence and almost sure convergence. J. Reine Angew. Math. 404 (1990), 140–161.Google Scholar
[Bo2]Bourgain, J.. On the maximal ergodic theorem for certain subsets of the integers. Israel J. Math. 61(1) (1988), 39–72.CrossRefGoogle Scholar
[F]Frantzikinakis, N.. Multiple recurrence and convergence for Hardy sequences of polynomial growth. J. Anal. Math. 112 (2010), 79–135.CrossRefGoogle Scholar
[FK]Frantzikinakis, N. and Kra, B.. Convergence of multiple ergodic averages for some commuting transformations. Ergod. Th. & Dynam. Sys. 25(3) (2005), 799–809.CrossRefGoogle Scholar
[H]Hopf, E.. Ergodentheorie. Springer, Berlin, 1937.CrossRefGoogle Scholar
[Ho1]Host, B.. Ergodic seminorms for commuting transformations and applications. Studia Math. 195 (2009), 31–49.CrossRefGoogle Scholar
[HoK1]Host, B. and Kra, B.. Non-conventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161(1) (2005), 397–488.CrossRefGoogle Scholar
[HoK2]Host, B. and Kra, B.. Convergence of polynomial ergodic averages. Israel J. Math. 149 (2005), 1–19.CrossRefGoogle Scholar
[J]Johnson, M.. Convergence of polynomial ergodic averages of several variables for some commuting transformations. Illinois J. Math. 53(3) (2009), 865–882.CrossRefGoogle Scholar
[Ko]Kolmogorov, A. N.. A simplified proof of the Birkhoff–Khinchin ergodic theorem. Uspekhi Mat. Nauk 5 (1938), 52–56.Google Scholar
[L1]Leibman, A.. Multiple recurrence theorem for measure preserving actions of a nilpotent group. Geom. Funct. Anal. 8 (1998), 853–931.CrossRefGoogle Scholar
[L2]Leibman, A.. Pointwise convergence of ergodic averages for polynomial actions of ℤd by translations on a nilmanifold. Ergod. Th. & Dynam. Sys. 25 (2005), 215–225.CrossRefGoogle Scholar
[L3]Leibman, A.. Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math. 146 (2005), 303–322.CrossRefGoogle Scholar
[L4]Leibman, A.. Rational sub-nilmanifolds of a compact nilmanifold. Ergod. Th. & Dynam. Sys. 26 (2006), 787–798.CrossRefGoogle Scholar
[L5]Leibman, A.. Orbits on a nilmanifold under the action of a polynomial sequence of translations. Ergod. Th. & Dynam. Sys. 27 (2007), 1239–1252.CrossRefGoogle Scholar
[vN]von Neumann, J.. Physical applications of the ergodic hypothesis. Proc. Natl. Acad. Sci. USA 18(3) (1932), 263–266.CrossRefGoogle Scholar
[P]Potts, A.. Multiple ergodic averages for flows and an application. Illinois J. Math. to appear. arXiv:0910.3687.Google Scholar
[Sh]Shah, N.. Limit distributions of polynomial trajectories on homogeneous spaces. Duke Math. J. 75(3) (1994), 711–732.CrossRefGoogle Scholar
[Z]Ziegler, T.. Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20(1) (2007), 53–97.CrossRefGoogle Scholar