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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 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 ergodicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case. Other… (More)

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

as i→∞.) A similar conjecture can be stated also in the finite volume case [6]. Of particular number theoretic interest are manifolds of the form Γ\H with Γ a congruence arithmetic lattice, in which case it is natural to assume that the eigenfunctions are Hecke-Maas forms, i.e. also eigenfunctions of all Hecke operators. We shall refer to this special 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.

- Elon Lindenstrauss
- Random Struct. Algorithms
- 1999

In this paper we give a counter example to a conjecture mentioned in Benjamini and Kesten (1996) about distinguishing sceneries by observing them along a simple random walk, giving an explicit construction of 2 @0 sceneries that are all pairwise indistinguishable.

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)

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)

- Elon Lindenstrauss, Y Set
- 1995

The main result we prove in this paper is that for any nite dimensional dynamical system (with topological entropy h), and for any factor with strictly lower entropy h 0 , there exist an intermediate factor of entropy h 00 for every h 00 2 h 0 ; h]. Two examples, one of them minimal, show that this is not the case for innnite dimensional systems. The… (More)