On the entropy of actions of nilpotent Lie groups and their lattice subgroups

Abstract

We consider a natural class $\mathcal {ULG}$ of connected, simply connected nilpotent Lie groups which contains ℝn, the group $\mathcal {UT}_n(\mathbb {R})$ of all triangular unipotent matrices over ℝ and many of its subgroups, and is closed under direct products. If $G \in \mathcal {ULG}$, then $\Gamma _1 = G\cap \mathcal {UT}_n(\mathbb {Z})$ is a lattice subgroup of G. We prove that if $G \in \mathcal {ULG}$ and Γ is a lattice subgroup of G, then a free ergodic measure-preserving action T of G on a probability space (X,ℬ,μ) has completely positive entropy (CPE) if and only if the restriction TΓ of T to Γ has CPE. We can deduce from this the following version of a well-known conjecture in this case: the action T has CPE if and only if T is uniformly mixing. Moreover, such T has a Lebesgue spectrum with infinite multiplicity. We further consider an ergodic free action T with positive entropy and suppose TΓ is ergodic for any lattice subgroup Γ of G. This holds, in particular, if the spectrum of T does not contain a discrete component. Then we show the Pinsker algebra Π(T) of T exists and coincides with the Pinsker algebras Π(TΓ) of TΓ for any lattice subgroup Γ of G. In this case, T always has Lebesgue spectrum with infinite multiplicity on the space ℒ20(X,μ)⊖ℒ20(Π(T)) , where ℒ20(Π(T)) contains all Π(T) -measurable functions from ℒ20(X,μ) . To prove these results, we use the following formula: h(T)=∣G(Γ)∣−1hK (TΓ) , where h(T) is the Ornstein–Weiss entropy of T, hK (TΓ) is a Kolmogorov–Sinai entropy of TΓ, and the number ∣G(TΓ)∣ is the Haar measure of the compact subset G(Γ) of G. In particular, h(T)=hK (TΓ1) , and hK (TΓ1)=∣G(Γ)∣−1hK (TΓ) . The last relation is an analogue of the Abramov formula for flows.