Related papers: Quantitative recurrence properties for piecewise e…

In this paper, we prove that if $S\subseteq\mathbb{R}^d$ is hyperplane absolute winning on a closed hyperplane diffuse set $L\subseteq\mathbb{R}^d$, then $\mathrm{dim}_H S\cap K=\mathrm{dim}_H K$ for any irreducible self-conformal set…

We give a fairly complete characterization of the exact components of a large class of uniformly expanding Markov maps of $\mathbb{R}$. Using this result, for a class of $\mathbb{Z}$-invariant maps and finite modifications thereof, we prove…

We consider two classes of piecewise expanding maps $T$ of $[0,1]$: a class of uniformly expanding maps for which the Perron-Frobenius operator has a spectral gap in the space of bounded variation functions, and a class of expanding maps…

We study the Bernoulli property for a class of partially hyperbolic systems arising from skew products. More precisely, we consider a hyperbolic map $(T,M,\mu)$, where $\mu$ is a Gibbs measure, an aperiodic H\"older continuous cocycle…

We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure $\mu$ with even continuous density and sections are of arbitrary dimension $n-k,\ 1\le k <n.$ If $K$ is a…

Let $S$ be a punctured Riemann surface with Euler characteristic $\chi(S)<0$. For any unitary representation $\rho: \pi_1(S) \to U(N)$, we introduce its renormalized energy and its harmonic representatives, which are equivariant harmonic…

Let $I\subset\mathbb{R}$ be an interval and $T_a:[0,1]\to[0,1]$, $a\in I$, a one-parameter family of piecewise expanding maps such that for each $a\in I$ the map $T_a$ admits a unique absolutely continuous invariant probability measure…

Building on advanced results on permutations, we show that it is possible to construct, for each irreducible representation of SU(N), an orthonormal basis labelled by the set of {\it standard Young tableaux} in which the matrix of the…

It is well-known that if T is a D_m-D_n bimodule map on the m by n complex matrices, then T is a Schur multiplier and $\|T\|_{cb}=\|T\|$. If n=2 and T is merely assumed to be a right D_2-module map, then we show that $\|T\|_{cb}=\|T\|$.…

Let $\lambda$ be a probability measure on $\mathbb T^{n-1}$ where $n=2$ or 3. Suppose $\lambda$ is invariant, ergodic and has positive entropy with respect to the linear transformation defined by a hyperbolic matrix. We get a measure $\mu $…

Given a parabolic cylinder $Q =(0,T)\times\Omega$, where $\Omega\subset \mathbb{R}^{N}$ is a bounded domain, we prove new properties of solutions of \[ u_t-\Delta_p u = \mu \quad \text{in $Q$} \] with Dirichlet boundary conditions, where…

We show the existence of Lebesgue-equivalent conservative and ergodic $\sigma$-finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures…

The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by…

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures $\mu$ in $n$-dimensional Euclidean space for all $n\geq 2$ in terms of…

A finitely-additive measure $\lambda $ on an infinite-dimensional real Hilbert space $E$ which is invariant with respect to shifts and orthogonal mappings has been defined. This measure can be considered as the analog of the Lebesgue…

Let X be a metric space with metric d and T,S be two commutative measure-preserving maps of X. In this paper we obtain numerical results about multiple recurrence of almost every point of this dynamical system. On other words we study the…

The $\mathit{growth\ rate\ function}$ for a nonempty minor-closed class of matroids $\mathcal{M}$ is the function $h_{\mathcal{M}}(n)$ whose value at an integer $n \ge 0$ is defined to be the maximum number of elements in a simple matroid…

In a previous paper the authors developed a H^1-BMO theory for unbounded metric measure spaces $(M,\rho,m)$ of infinite measure that are locally doubling and satisfy two geometric properties, called "approximate midpoint" property and…

Let $(X, \mathcal{B},\mu,T)$ be an ergodic measure preserving system, $A \in \mathcal{B}$ and $\epsilon>0$. We study the largeness of sets of the form \begin{equation*} \begin{split} S = \left\{ n\in\mathbb{N}\colon\mu(A\cap…

Akcoglu and Suchaston proved the following result: Let $T:L^1(X,{\cf},\m)\to L^1(X,{\cf},\m)$ be a positive contraction. Assume that for $z\in L^1(X,{\cf},\m)$ the sequence $(T^nz)$ converges weakly in $L^1(X,{\cf},\m)$, then either…