Directional recurrence and directional rigidity for infinite measure preserving actions of nilpotent lattices

Abstract

Let $\Gamma$ be a lattice in a simply connected nilpotent Lie group $G$. Given an infinite measure preserving action $T$ of $\Gamma$ and a "direction" in $G$ (i.e. an element $\theta$ of the projective space $P(\goth g)$ of the Lie algebra $\goth g$ of $G$), some notions of recurrence and rigidity for $T$ along $\theta$ are introduced. It is shown that the set of recurrent directions $\Cal R(T)$ and the set of rigid directions for $T$ are both $G_\delta$. In the case where $G=\Bbb R^d$ and $\Gamma=\Bbb Z^d$, we prove that (a) for each $G_\delta$-subset $\Delta$ of $P(\goth g)$ and a countable subset $D\subset\Delta$, there is a rank-one action $T$ such that $D\subset\Cal R(T)\subset\Delta$ and (b) $\Cal R(T)=P(\goth g)$ for a generic infinite measure preserving action $T$ of $\Gamma$. This answers partly a question from a recent paper by A.~Johnson and A.~{\c S}ahin. Some applications to the directional entropy of Poisson actions are discussed. In the case where $G$ is the Heisenberg group $H_3(\Bbb R)$ and $\Gamma=H_3(\Bbb Z)$, a rank-one $\Gamma$-action $T$ is constructed for which $\Cal R(T)$ is not invariant under the natural "adjoint" $G$-action.