Directional ergodicity and weak mixing for actions of ${\mathbb{R}}^d$ and ${\mathbb{Z}}^d$

E. Arthur Robinson, Joseph Rosenblatt, Ayşe A. Şahin

We define notions of direction $L$ ergodicity, weak mixing, and mixing for a measure preserving ${\mathbb{Z}}^d$ action $T$ on a Lebesgue probability space $(X,\mu )$, where $L\subseteq\…