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We study a random map $T$ which consists of intermittent maps $\{T_{k}\}_{k=1}^{K}$ and a position dependent probability distribution $\{p_{k,\varepsilon}(x)\}_{k=1}^{K}$. We prove existence of a unique absolutely continuous invariant…

We prove a fiberwise almost sure invariance principle for random piecewise expanding transformations in one and higher dimensions using recent developments on martingale techniques.

We introduce a parametrised family of maps $\{S_{\eta}\}_{\eta \in [1,2]}$, called symmetric doubling maps, defined on $[-1,1]$ by $S_\eta (x)=2x-d\eta$, where $d\in \{-1,0,1 \}$. Each map $S_\eta$ generates binary expansions with digits…

A sufficient geometrical condition for the existence of absolutely continuous invariant probability measures for S-unimodal maps will be discussed. The Lebesgue typical existence of such measures in the quadratic family will be a…

We consider expanding systems with invariant measures that are uniformly expanding everywhere except on a small measure set and show that the limiting statistics of hitting times for zero measure sets are compound Poisson provided the…

We consider perturbations of quadratic maps $f_a$ admitting an absolutely continuous invariant probability measure, where $a$ is in a certain positive measure set $\mathcal{A}$ of parameters, and show that in any neighborhood of any such an…

Given $d\ge 1$, we provide a construction of the random measure - the critical Gaussian Multiplicative Chaos - formally defined $e^{\sqrt{2d}X}\mathrm{d} \mu$ where $X$ is a $\log$-correlated Gaussian field and $\mu$ is a locally finite…

We utilize Gaussian measure preserving systems to prove the existence and genericity of Lebesgue measure preserving transformations $T:[0,1]\rightarrow [0,1]$ which exhibit both mixing and rigidity behavior along families of asymptotically…

Consider the random matrix $\Sigma = D^{1/2} X \widetilde D^{1/2}$ where $D$ and $\widetilde D$ are deterministic Hermitian nonnegative matrices with respective dimensions $N \times N$ and $n \times n$, and where $X$ is a random matrix with…

We consider a family of Pomeau-Manneville type interval maps $T_\alpha$, parametrized by $\alpha \in (0,1)$, with the unique absolutely continuous invariant probability measures $\nu_\alpha$, and rate of correlations decay $n^{1-1/\alpha}$.…

We consider densities $D_\Sigma(A)$, $\overline{D}_\Sigma(A)$ and $\underline{D}_\Sigma(A)$ for a subset $A$ of $\mathbb{N}$ with respect to a sequence $\Sigma$ of finite subsets of $\mathbb{N}$ and study Fourier coefficients of ergodic,…

We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an $\ell^p$ condition and a generalized bounded variation condition. This latter condition requires that a sequence can be…

For each integer $n\ge 1$, denote by $T_{n}$ the map $x\mapsto nx\mod 1$ from the circle group $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ into itself. Let $p,q\ge 2$ be two multiplicatively independent integers. Using Baire Category arguments, we…

We consider a piecewise smooth expanding map of the interval possessing two invariant subsets of positive Lebesgue measure and exactly two ergodic absolutely continuous invariant probability measures (ACIMs). When this system is perturbed…

Let $(X,\mathcal{A}, \mu)$ be a probability measure space and let $T_i,$ $1\leq i\leq H,$ be commuting invertible measure preserving transformations on this measure space. We prove the following pointwise results; The averages…

This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures $\mu_i$ and $\nu_i, i=1,2$, are close to each other in terms of the L\'{e}vy metric and if the…

Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers $p_{1},\dots,p_{r}$ there exists a continuous probability measure $\mu $ on the unit circle…

We prove that for certain partially hyperbolic skew-products, non-uniform hyperbolicity along the leaves implies existence of a finite number of ergodic absolutely continuous invariant probability measures which describe the asymptotics of…

We analyze certain parametrized families of one-dimensional maps with infinitely many critical points from the measure-theoretical point of view. We prove that such families have absolutely continuous invariant probability measures for a…

We study some special classes of piecewise continuous maps on a finite smooth partition of a compact manifold and look for invariant measures for such maps. We show that in the simplest one-dimensional case (so-called interval translation…