Related papers: Extremality and dynamically defined measures, part…

We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and…

Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss, we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has…

Let $\Lambda$ be the limit set of a conformal dynamical system, i.e. a Kleinian group acting on either finite- or infinite-dimensional real Hilbert space, a conformal iterated function system, or a rational function. We give an easily…

This article connects the theory of extremal doubly stochastic measures to the geometry and topology of optimal transportation. We begin by reviewing an old question (# 111) of Birkhoff in probability and statistics [4], which is to give a…

We prove that every hyperbolic measure invariant under a C^{1+\alpha} diffeomorphism of a smooth Riemannian manifold possesses asymptotically ``almost'' local product structure, i.e., its density can be approximated by the product of the…

We study the finiteness of physical measures for skew-product transformations $F$ associated with discrete-time random dynamical systems driven by ergodic Markov chains. We develop a framework, using an independent and identically…

This paper is partly an exposition, and partly an extension of our work [1] to the multiparameter case. We consider certain classes of parametrized dynamically defined measures. These are push-forwards, under the natural projection, of…

We study the structure of invariant measures for continuous automorphisms of compact metrizable abelian groups satisfying the descending chain condition. We show that the finitely supported invariant measures are weak-* dense in the space…

We give a new, two-step approach to prove existence of finite invariant measures for a given Markovian semigroup. First, we identify a convenient auxiliary measure and then we prove conditions equivalent to the existence of an invariant…

The multifractal formalism for measures in its original formulation is checked for special classes of measures such as doubling, self-similar, and Gibbs-like ones. Out of these classes, suitable conditions should be taken into account to…

We study continuity and discontinuity properties of some popular measure-dimension mappings under some topologies on the space of probability measures in this work. We give examples to show that no continuity can be guaranteed under general…

We investigate the supports of extremal martingale measures with pre-specified marginals in a two-period setting. First, we establish in full generality the equivalence between the extremality of a given measure $Q$ and the denseness in…

In this paper we prove a conjecture of Kleinbock and Tomanov \cite[Conjecture~FP]{KT} on Diophantine properties of a large class of fractal measures on $\mathbb{Q}_p^n$. More generally, we establish the $p$-adic analogues of the influential…

The goal of this paper is to generalize the main results of [KM] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish `joint strong…

We solve the problem of giving sharp asymptotic bounds on the Hausdorff dimensions of certain sets of badly approximable matrices, thus improving results of Broderick and Kleinbock (preprint 2013) as well as Weil (preprint 2013), and…

We introduce and study skew product Smale endomorphisms over finitely irreducible topological Markov shifts with countable alphabets. We prove that almost all conditional measures of equilibrium states of summable and locally Holder…

Let $W(\p)$ denote the set of $\p$-well approximable points in $\R^d$ and let $K$ be a compact subset of $\R^d$ which supports a measure $\mu$. In this short note, we show that if $\mu$ is an `absolutely friendly' measure and a certain…

We prove the finiteness of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms where the center direction has a dominated decomposition into one dimensional bundle and there is a uniform lower bound for the absolute…

In a variety of applications it is important to extract information from a probability measure $\mu$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and possibly conditioned) continuous time…

We establish a new connection between metric Diophantine approximation and the parametric geometry of numbers by proving a variational principle facilitating the computation of the Hausdorff and packing dimensions of many sets of interest…