相关论文: Measure rigidity and equidistribution for fractal …
We provide a new definition of a local walk dimension $\beta$ that depends only on the metric and not on the existence of a particular regular Dirichlet form or heat kernel asymptotics. Moreover, we study the local Hausdorff dimension…
We analyse how simple local constraints in two dimensions lead a defect to exhibit robust, non-transient, and tunable, subdiffusion. We uncover a rich dynamical phenomenology realised in ice- and dimer-type models. On the microscopic scale…
The spreading properties of the stationary states of the quantum multidimensional harmonic oscillator are analytically discussed by means of the main dispersion measures (radial expectation values) and the fundamental entropy-like…
We consider measures which are invariant under a measurable iterated function system with positive, place-dependent probabilities in a separable metric space. We provide an upper bound of the Hausdorff dimension of such a measure if it is…
We study products of random isometries acting on Euclidean space. Building on previous work of the second author, we prove a local limit theorem for balls of shrinking radius with exponential speed under the assumption that a Markov…
We study the bi-Lipschitz classification of Bedford-McMullen carpets which are totally disconnected. Let $E$ be a such carpet and let $\mu_E$ be the uniform Bernoulli measure on $E$. We show that the multifractal spectrum and the doubling…
Based on the seminal work of Hutchinson, we investigate properties of {\em $\alpha$-weighted Cantor measures} whose support is a fractal contained in the unit interval. Here, $\alpha$ is a vector of nonnegative weights summing to $1$, and…
Let $\{x\_n\}\_{n\geq 0}$ be a sequence of $[0,1]^d$, $\{\lambda\_n\} \_{n\geq 0}$ a sequence of positive real numbers converging to 0, and $\delta>1$. Let $\mu$ be a positive Borel measure on $[0,1]^d$, $\rho\in (0,1]$ and $\alpha>0$.…
We study properties of Bernoulli convolutions generated by the second Ostrogradsky series, i.e., probability distributions of random variables \begin{equation} \xi = \sum_{k=1}^\infty \frac{(-1)^{k+1}\xi_k}{q_k}, \end{equation} where $q_k$…
The contribution of this work is twofold. The first part deals with a Hilbert-space version of McCann's celebrated result on the existence and uniqueness of monotone measure-preserving maps: given two probability measures $\rm P$ and $\rm…
This document offers a concise introduction to the mathematical theory and practical application of the Hausdorff Measure and Dimension. The primary objective is to clarify and rigorously detail the two most common methods used for…
We introduce a randomized iterative fragmentation procedure for finite metric spaces, which is guaranteed to result in a polynomially large subset that is $D$-equivalent to an ultrametric, where $D\in (2,\infty)$ is a prescribed target…
We prove that the dimension drop phenomenon holds for the harmonic measure associated to a transient random walk in a random environment (as defined by R. Lyons and R. Pemantle in 1992 and generalized by G. Faraud in 2011) on an infinite…
Let $w\in L^1\_{loc}(\R^n)$ be apositive weight. Assuming that a doubling condition and an $L^1$ Poincar\'e inequality on balls for the measure $w(x)dx$, as well as a growth condition on $w$, we prove that the compact subsets of $\R^n$…
A one-dimensional confined Nonlinear Random Walk is a tuple of $N$ diffeomorphisms of the unit interval driven by a probabilistic Markov chain. For generic such walks, we obtain a geometric characterization of their ergodic stationary…
We consider invariant transports of stationary random measures on $\mathbb{R}^d$ and establish natural mixing criteria that guarantee persistence of asymptotic variances. To check our mixing assumptions, which are based on two-point Palm…
We investigate ergodic-theoretical quantities and large deviation properties of one-dimensional intermittent maps, that have not only an indifferent fixed point but also a singular structure such that the uniform measure is invariant under…
We consider a random walk on a closed manifold $M$ driven by a probability measure $\mu$ on the space of $C^2$ diffeomorphisms. Provided $\mu$ has compact support, satisfies certain gap and pinching conditions, and is weak-$*$ close to a…
We prove an equidistribution theorem a la Bader-Muchnik for operator-valued measures associated with boundary representations in the context of discrete groups of isometries of CAT(-1) spaces thanks to an equidistribution theorem of T.…
We give new examples and describe the complete lists of all measures on the set of countable homogeneous universal graphs and $K_s$-free homogeneous universal graphs (for $s\geq 3$) that are invariant with respect to the group of all…