Multiple and Polynomial Recurrence for Abelian Actions in Infinite Measure

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

1. a p-recurrent actionT=(T_g)_gG such that (if p) no one transformationT_g is (p+1)-recurrentfor every element g of infinite order;
2. an action T=(T_g)_gGsuch that for every finite sequence g1,...,g_rGwithout torsionthe transformation T_g1x...x T_gr is ergodic,p-recurrent but(if p) not (p+1)-recurrent;
3. a p-polynomially recurrent (C,F)-transformationwhich (if p)is not (p+1)-recurrent.

-recurrence here meansmultiple recurrence. Moreover, it is shown that there existsa (C,F)-transformation which is rigid (and hence multiply recurrent)but not polynomially recurrent. Nevertheless, the subset ofpolynomially recurrent transformations is generic in the groupof infinite measure preserving transformations endowed withthe weak topology.