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

Abstract

Let$\unicode[STIX]{x1D6E4}$be a lattice in a simply connected nilpotent Lie group$G$. Given an infinite measure-preserving action$T$of$\unicode[STIX]{x1D6E4}$and a ‘direction’ in$G$(i.e. an element$\unicode[STIX]{x1D703}$of the projective space$P(\mathfrak{g})$of the Lie algebra$\mathfrak{g}$of$G$), some notions of recurrence and rigidity for$T$along$\unicode[STIX]{x1D703}$are introduced. It is shown that the set of recurrent directions${\mathcal{R}}(T)$and the set of rigid directions for$T$are both$G_{\unicode[STIX]{x1D6FF}}$. In the case where$G=\mathbb{R}^{d}$and$\unicode[STIX]{x1D6E4}=\mathbb{Z}^{d}$, we prove that (a) for each$G_{\unicode[STIX]{x1D6FF}}$-subset$\unicode[STIX]{x1D6E5}$of$P(\mathfrak{g})$and a countable subset$D\subset \unicode[STIX]{x1D6E5}$, there is a rank-one action$T$such that$D\subset {\mathcal{R}}(T)\subset \unicode[STIX]{x1D6E5}$and (b)${\mathcal{R}}(T)=P(\mathfrak{g})$for a generic infinite measure-preserving action$T$of$\unicode[STIX]{x1D6E4}$. This partly answers a question from a recent paper by Johnson and Şahin. Some applications to the directional entropy of Poisson actions are discussed. In the case where$G$is the Heisenberg group$H_{3}(\mathbb{R})$and$\unicode[STIX]{x1D6E4}=H_{3}(\mathbb{Z})$, a rank-one$\unicode[STIX]{x1D6E4}$-action$T$is constructed for which${\mathcal{R}}(T)$is not invariant under the natural ‘adjoint’$G$-action.