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

Let $Γ$ be a lattice in a simply connected nilpotent Lie group $G$. Given an infinite measure preserving action $T$ of $Γ$ and a "direction" in $G$ (i.e. an element $θ$ of the projective space $P(\goth g)$ of the Lie algebra $\goth g$ of $G$), some notions of recurrence and rigidity for $T$ along $θ$ are introduced. It is shown that the set of recurrent directions $\Cal R(T)$ and the set of rigid directions for $T$ are both $G_δ$. In the case where $G=\Bbb R^d$ and $Γ=\Bbb Z^d$, we prove that (a) for each $G_δ$-subset $Δ$ of $P(\goth g)$ and a countable subset $D\subsetΔ$, there is a rank-one action $T$ such that $D\subset\Cal R(T)\subsetΔ$ and (b) $\Cal R(T)=P(\goth g)$ for a generic infinite measure preserving action $T$ of $Γ$. 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 $Γ=H_3(\Bbb Z)$, a rank-one $Γ$-action $T$ is constructed for which $\Cal R(T)$ is not invariant under the natural "adjoint" $G$-action.

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