Elon Lindenstrauss

Pointwise theorems for amenable groups

Abstract.In this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along Følner sequences that obey some restrictions. These restrictions are mild enough so that such sequences exist for all amenable groups. We also prove a generalization of the Shannon-McMillan-Breiman theorem to all discrete amenable groups. -->

Mean topological dimension

In this paper we present some results and applications of a new invariant for dynamical systems that can be viewed as a dynamical analogue of topological dimension. This invariant has been introduced by M. Gromov, and enables one to assign a meaningful quantity to dynamical systems of infinite topological dimension and entropy. We also develop an alternative approach that is metric dependent and is intimately related to topological entropy.

Mean dimension, small entropy factors and an embedding theorem

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 factors with arbitrarily small topological entropy, and when can a system (X, T) be embedded in (([0, 1]d)Z, shift). Our results apply to extensions of minimalZ-actions, and for this case we also show that there is a very satisfying dimension theory for mean dimension.

Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus

Let Γ be a semigroup of d × d nonsingular integer matrices, and consider the action of Γ on the torus T. We assume throughout that the action is strongly irreducible: there is no subtorus invariant under a finite index subsemigroup of Γ. The strong irreducibility assumption in particular implies that Γ acts ergodically on T (equipped with the Lebesgue measure m). Therefore the Γ-orbit of Lebesgue almost every x ∈ T is dense and in an appropriate sense even becomes equidistributed. However, when Γ is cyclic, there is a set of full Hausdorff dimension of exceptional points x for which Γ.x fails to be dense. When Γ is bigger, the distribution of individual Γ-orbits can be expected to be much more restrictive. An important result in this direction is due to Furstenberg, who showed for d = 1 (in which case Γ 1 an abelian semigroup never satisfies condition (Γ-1); indeed, the group generated by a semigroup satisfying (Γ-1) is nonamenable. Assumption (Γ-2) is a technical condition which is in particular satisfied when Γ is a Zariski dense semigroup of SLd(Z) [16].

On sets invariant under the action of the diagonal group

We consider the action of the (n-1) -dimensional group of diagonal matrices in SL (n,\mathbb{R}) on SL (n,\mathbb{R})/\Gamma , where \Gamma is a lattice and n\ge 3 . Far-reaching conjectures of Furstenberg, Katok–Spatzier and Margulis suggest that there are very few closed invariant sets for this action. We examine the closed invariant sets containing compact orbits. For example, for \Gamma={\rm SL}(n,\mathbb{Z}) we describe all possible orbit-closures containing a compact orbit. In marked contrast to the case n=2 , such orbit-closures are necessarily homogeneous submanifolds in the sense of Ratner.

Distribution of periodic torus orbits on homogeneous spaces

We prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of PGLn(R). After attaching to each periodic orbit an integral invariant (the discriminant) our results have the following flavour: certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most O(� ǫ ) orbits of discriminant ≤ �. The proof relies on the well-separatedness of periodic orbits together with measure rigidity for torus actions. We also give examples of sequences of periodic orbits of this action that fail to become equidistributed, even in higher rank. We give an application of our results to sharpen a theorem of Minkowski on ideal classes in totally real number fields of cubic and higher degrees.

Rigidity properties of \zd-actions on tori and solenoids

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 Zd-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 nonirreducible case. These results have applications regarding measurable factors and joinings of these algebraic Zd-actions.

Distribution of periodic torus orbits and Duke"s theorem for cubic fields

We study periodic torus orbits on spaces of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits, and show that the equivalence classes become uniformly distributed. This is a cubic analogue of Dukes theorem about the distribution of closed geodesics on the modular surface: suitably interpreted, the ideal classes of a cubic totally real field are equidistributed in the modular 5-fold SL_3(Z)\SL_3(R)/SO_3(R). In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal compact flats in SL_3(Z)\SL_3(R)/SO_3(R) of volume less than V becomes equidistributed as V goes to infinity. The proof combines subconvexity estimates, measure classification, and local harmonic analysis.

On quantum unique ergodicity for Γ \ ℍ × ℍ

The purpose of this note is to point out a connection between the Quantum Unique Ergodicity conjecture of Z. Rudnick and P. Sarnak (or, more precisely, natural higher rank generalizations of this conjecture) and conjectures of H. Furstenberg, A. Katok, and R. Spatzier, and G. Margulis regarding the scarcity of measures invariant under natural ℝ d actions (d ≥ 2) in the ergodic theory of Lie groups. Our main tool is a new variant of the micro local lift of A. Schnirelman,Y. Colin de Verdiere, and S. Zelditch with additional invariance properties. We also sharpen and generalize a related result of Rudnick and Sarnak on scarring of Hecke eigenforms.

Lowering topological entropy

The main result we prove in this paper is that for any finite dimensional dynamical system (with topological entropyh), and for any factor with strictly lower entropyh′, there exists an intermediate factor of entropyh″ for everyh″∈[h′, h]. Two examples, one of them minimal, show that this is not the case for infinite dimensional systems.