arXiv subjects

Yiftach Dayan

Publications and source records attributed to Yiftach Dayan.

4 recordsLinked to original sources

arXiv2022

Random walks on tori and normal numbers in self similar sets

We study random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. We deduce that if $K \subset \mathbb{R}^d$ is an attractor of a finite iterated function system of $n\geq 2$ maps of the form $x \mapsto D^{-r_i} x + t_i \ (i=1, \ldots, n)$, where $D$ is an expanding $d\times d$ integer matrix, and is the same for all the maps, and $r_{i} \in\mathbb{N}$, under an irrationality condition on the translation parts $t_i$, almost every point in $K$ (w.r.t. any Bernoulli measure) has an equidistributed orbit under the map $x\mapsto Dx$ (multiplication mod $\mathbb{Z}^{d}$). In the one-dimensional case, this conclusion amounts to normality to base $D$. Thus for example, almost every point in an irrational dilation of the middle-thirds Cantor set is normal to base 3.

math.DS↗

arXiv2022

On microsets, Assouad dimension and lower dimension of random fractals, and Furstenberg's homogeneity

We study the collection of microsets of randomly constructed fractals, which in this paper, are referred to as Galton-Watson fractals. This is a model that generalizes Mandelbrot percolation, where Galton-Watson trees (whose offspring distribution is not necessarily binomial) are projected to $\mathbb{R}^d$ by a coding map which arises from an iterated function system (IFS) of similarity maps. We show that for such a random fractal $E$, whenever the underlying IFS satisfies the open set condition, almost surely the Assouad dimension of $E$ is the maximal Hausdorff dimension of a set in $\text{supp}\left(E\right)$, the lower dimension is the smallest Hausdorff dimension of a set in $\text{supp}\left(E\right)$, and every value in between is the Hausdorff dimension of some microset of $E$. In order to obtain the above, we first analyze the relation between the collection of microsets of a (deterministic) set, and certain limits of subtrees of an appropriate coding tree for that set. The results of this analysis are also applied, with the required adjustments, to gain some insights on Furstenberg's homogeneity property. We define a weaker property than homogeneity and show that for self-homothetic sets in $\mathbb{R}$ whose Hausdorff dimension is smaller than 1, it is equivalent to the weak separation condition.

math.DS↗

arXiv2019

Diophantine approximations on random fractals

We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realization of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$, $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$, where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$. We show this by proving that $E$ almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$. We achieve this by analyzing Galton-Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of $E$. This method allows us to obtain a more general result by projecting the Galton-Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.

math.PR↗

arXiv2013

Hypercyclic operators, Gauss measures and Polish dynamical systems

In this work we consider hypercyclic operators as a special case of Polish dynamical systems. In the first section we analyze the construction of Bayart and Grivaux of a hypercyclic operator which preserves a Gaussian measure, and derive a description of the maximal spectral type of the Koopman operator associated to the corresponding measure preserving dynamical system. We then use this information to show the existence of a mildly but not strongly mixing hypercyclic operator on Hilbert space. In the last two sections we study hypercyclic and frequently hypecyclic operators which, as Polish dynamical systems are, M-systems, E-systems, and syndetically transitive systems.

math.DS↗