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We classify the measures on SL(k, R)/ SL(k, Z) which are invariant and ergodic under the action of the group A of positive diagonal matrices with positive entropy. We apply this to prove that the set of exceptions to Littlewood's conjecture has Hausdorff dimension zero.

We classify measures on the locally homogeneous space Γ\ SL(2, R) × L which are invariant and have positive entropy under the diagonal subgroup of SL(2, R) and recurrent under L. This classification can be used to show arithmetic quantum unique er-godicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case.… (More)

In this paper we describe proofs of the pointwise ergodic theorem and Shannon-McMillan-Breiman theorem for discrete amenable groups, along Følner sequences that obey some restrictions. These restrictions are mild enough so that such sequences exist for all amenable groups.

In this paper we present some results and applications of a new invariant for dynamical systems that can be viewed as a dy-namical analogue of topological dimension. This invariant has been introduced by M. Gromov, and enables one to assign a meaningful quantity to dynamical systems of innnite topological dimension and entropy. We also develop an… (More)

We study diophantine properties of a typical point with respect to measures on R n. Namely, we identify geometric conditions on a measure µ on R n guaranteeing that µ-almost every y ∈ R n is not very well multi-plicatively approximable by rationals. Measures satisfying our conditions are called 'friendly'. Examples include smooth measures on nondegenerate… (More)

In this paper we show how the notion of mean dimension is connected in a natural way to the following two questions: what points in a dynamical system (X; T) can be distinguished by factor with arbitrarily small topological entropy, and when can a system (X; T) be imbedded in ? ((0; 1] d) Z ; shift. Our results apply to extensions of minimal Z-actions, and… (More)

Let G be a split adjoint semisimple group over Q and K∞ ⊂ G(R) a maximal compact subgroup. We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of G(R)/K∞. This proves a conjecture of Sarnak for Q-split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type… (More)

Given ergodic p-invariant measures f i g on the 1-torus T = R=Z, we give a sharp condition on their entropies, guaranteeing that the entropy of the convolution 1 n converges to log p. We also prove a variant of this result for joinings of full entropy on T N. In conjunction with a method of Host, this yields the following. Denote q (x) = qx (mod 1). Then… (More)

We show that Haar measure is a unique measure on a torus or more generally a solenoid X invariant under a not virtually cyclic totally irreducible Z d-action by automorphisms of X such that at least one element of the action acts with positive entropy. We also give a corresponding theorem in the non-irreducible case. These results have applications… (More)