Invariant measures and the set of exceptions to Littlewood's conjecture

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.Comment: 48 page

On the critical points of the entropic principle

In a recent paper, hep-th/0509109, Gukov et al. introduced an entropy functional on the moduli space of Calabi-Yau compactifications. The maxima of this functional are then interpreted as "preferred" Calabi-Yau compactifications. In this note we show that for compact Calabi-Yaus, all regular critical points of this entropic principle are maxima.Comment: 11 page

Topological Entropy and Algebraic Entropy for group endomorphisms

The notion of entropy appears in many fields and this paper is a survey about entropies in several branches of Mathematics. We are mainly concerned with the topological and the algebraic entropy in the context of continuous endomorphisms of locally compact groups, paying special attention to the case of compact and discrete groups respectively. The basic properties of these entropies, as well as many examples, are recalled. Also new entropy functions are proposed, as well as generalizations of several known definitions and results. Furthermore we give some connections with other topics in Mathematics as Mahler measure and Lehmer Problem from Number Theory, and the growth rate of groups and Milnor Problem from Geometric Group Theory. Most of the results are covered by complete proofs or references to appropriate sources

Generic Stationary Measures and Actions

Let $G$ be a countably infinite group, and let $\mu$ be a generating probability measure on $G$. We study the space of $\mu$-stationary Borel probability measures on a topological $G$ space, and in particular on $Z^G$, where $Z$ is any perfect Polish space. We also study the space of $\mu$-stationary, measurable $G$-actions on a standard, nonatomic probability space. Equip the space of stationary measures with the weak* topology. When $\mu$ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of $(G,\mu)$. When $Z$ is compact, this implies that the simplex of $\mu$-stationary measures on $Z^G$ is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on $\{0,1\}^G$. We furthermore show that if the action of $G$ on its Poisson boundary is essentially free then a generic measure is isomorphic to the Poisson boundary. Next, we consider the space of stationary actions, equipped with a standard topology known as the weak topology. Here we show that when $G$ has property (T), the ergodic actions are meager. We also construct a group $G$ without property (T) such that the ergodic actions are not dense, for some $\mu$. Finally, for a weaker topology on the set of actions, which we call the very weak topology, we show that a dynamical property (e.g., ergodicity) is topologically generic if and only if it is generic in the space of measures. There we also show a Glasner-King type 0-1 law stating that every dynamical property is either meager or residual.Comment: To appear in the Transactions of the AMS, 49 page