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

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Journal Article
Ergodic Theory and Dynamical Systems, 2012, 32 (2), pp. 535 - 573
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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) -1hK (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) -1hK (T γ). The last relation is an analogue of the Abramov formula for flows. © 2011 Cambridge University Press.
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