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

TL;DR

This paper investigates recurrence and rigidity in infinite measure-preserving actions of nilpotent lattices, establishing generic properties and constructing specific actions with prescribed directional behaviors, with applications to entropy and non-invariance phenomena.

Contribution

It introduces notions of directional recurrence and rigidity for such actions, proves their genericity, and constructs examples with specific directional properties, partly answering open questions.

Findings

Recurrent and rigid directions form $G_\delta$ sets.

Existence of rank-one actions with prescribed directional recurrence sets.

Construction of actions with non-invariant directional recurrence sets in the Heisenberg group.

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.