Return times, recurrence densities and entropy for actions of some discrete amenable groups

Michael Hochman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

It is a theorem of Wyner and Ziv and Ornstein and Weiss that if one observes the initial k symbols X0,..., Xk-1 of a typical realization of a finite valued ergodic process with entropy h, the waiting time until this sequence appears again in the same realization grows asymptotically like 2hk [7, 12]. A similar result for random fields was obtained in [8]: in this case, one observes cubes in ℤd instead of initial segments. In the present paper, we describe generalizations of this. We examine what happens when the set of possible return times is restricted. Fix an increasing sequence of sets of possible times {Wn }and define R k to be the first n such that X0,..., Xk-1 recurs at some time in Wn. It turns out that |WRk| cannot drop below 2hk asymptotically. We obtain conditions on the sequence {Wn} which ensure that \WRk| is asymptotically equal to 2hk. We consider also recurrence densities of initial blocks and derive a uniform Shannon-McMillan-Breiman theorem. Informally, if U k,n is the density of recurrences of the block X0,..., Xk-1 in X-n,...,Xn, then Uk,n grows at a rate of 2hk, uniformly in n. We examine the conditions under which this is true when the recurrence times are again restricted to some sequence of sets {Wn} The above questions are examined in the general context of finite-valued processes parametrized by discrete amenable groups. We show that many classes of groups have time-sequences {Wn} along which return times and recurrence densities behave as expected. An interesting feature here is that this can happen also when the time sequence lies in a small subgroup of the parameter group.

Original languageAmerican English
Pages (from-to)1-51
Number of pages51
JournalJournal d'Analyse Mathematique
Volume100
DOIs
StatePublished - 2006

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