MULTIPLE AND POLYNOMIAL RECURRENCE FOR ABELIAN ACTIONS IN INFINITE MEASURE

Abstract

The $(C,F)$-construction from a previous paper of the first author is applied to produce a number of funny rank one infinite measure preserving actions of discrete countable Abelian groups $G$ with ‘unusual’ multiple recurrence properties. In particular, the following are constructed for each $p\in\Bbb N\cup\{\infty\}$:

1. (i) a $p$-recurrent action $T=(T_g)_{g\in G}$ such that (if $p\ne\infty$) no one transformation $T_g$ is $(p+1)$-recurrent for every element $g$ of infinite order;

2. (ii) an action $T=(T_g)_{g\in G}$ such that for every finite sequence $g_1,\dots,g_r\in G$ without torsion the transformation $T_{g_1}\times\cdots\times T_{g_r}$ is ergodic, $p$-recurrent but (if $p\ne\infty$) not $(p+1)$-recurrent;

3. (iii) a $p$-polynomially recurrent $(C,F)$-transformation which (if $p\ne\infty$) is not $(p+1)$-recurrent.

$\infty$-recurrence here means multiple recurrence. Moreover, it is shown that there exists a $(C,F)$-transformation which is rigid (and hence multiply recurrent) but not polynomially recurrent. Nevertheless, the subset of polynomially recurrent transformations is generic in the group of infinite measure preserving transformations endowed with the weak topology.