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

It is a theorem of Wyner and Ziv and Ornstein and Weiss that if one observes the initial k symbols X₀,..., X_k-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 2^hk [7, 12]. A similar result for random fields was obtained in [8]: in this case, one observes cubes in ℤ^d 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 {W_n }and define R _k to be the first n such that X₀,..., X_k-1 recurs at some time in W_n. It turns out that |W_Rk| cannot drop below 2^hk asymptotically. We obtain conditions on the sequence {W_n} which ensure that \W_Rk| is asymptotically equal to 2^hk. 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,..., X_k-1 in X_-n,...,X_n, then U_k,n grows at a rate of 2^hk, uniformly in n. We examine the conditions under which this is true when the recurrence times are again restricted to some sequence of sets {W_n} 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 {W_n} 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.