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

摘要

We consider a natural class ULG of connected, simply connected nilpotent Lie groups which contains R ~n, the group UT _nR of all triangular unipotent matrices over R and many of its subgroups, and is closed under direct products. If GULG, then Γ _1 = GUT _nZ is a lattice subgroup of G. We prove that if GULG and is a lattice subgroup of G, then a free ergodic measure-preserving action T of G on a probability space (X,β,μ) has completely positive entropy (CPE) if and only if the restriction T ~γ 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 T~ γ 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 (T ~γ) of T ~γ for any lattice subgroup γ of G. In this case, T always has Lebesgue spectrum with infinite multiplicity on the space L ~20(X,μ)θ-L ~20(π (T)), where L ~20(π (T)) contains all π (T)-measurable functions from L ~20(X,μ). To prove these results, we use the following formula: h(T)=G(γ) ~(-1)hK (T ~γ), where h(T) is the Ornstein-Weiss entropy of T, hK (T ~γ) is a Kolmogorov-Sinai entropy of T ~γ, and the number G(T ~γ) is the Haar measure of the compact subset G(γ) of G. In particular, h(T)=hK (T ~(γ1)), and hK (T ~(γ1))=G(γ) ~(-1)hK (T ~γ). The last relation is an analogue of the Abramov formula for flows.