Related papers: Hausdorff dimension of recurrence sets

We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set "cancels randomness" in the sense that some of its members, when added to Martin-Lof random reals, identify a point with lower…

In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable…

Suppose $n\geq 2$ and $\mathcal{A}_{i}\subset \{0,1,\cdots ,(n-1)\}$ for $ i=1,\cdots ,l,$ let $K_{i}=\bigcup\nolimits_{a\in \mathcal{A}_{i}}n^{-1}(K_{i}+a)$ be self-similar sets contained in $[0,1].$ Given $ m_{1},\cdots ,m_{l}\in…

Let $f$ be a homeomorphism of the closed annulus $A$ isotopic to the identity, and let $X\subset {\rm Int}A$ be an $f$-invariant continuum which separates $A$ into two domains, the upper domain $U_+$ and the lower domain $U_-$. Fixing a…

In this paper we implement a numerical algorithm to compute codimension-one tori in three-dimensional, volume-preserving maps. A torus is defined by its conjugacy to rigid rotation, which is in turn given by its Fourier series. The…

In this paper, we investigate the dimension theory of the one parameter family of Okamoto's function. We compute the Hausdorff, box-counting and Assouad dimensions of the graph for a typical choice of parameter. Furthermore, we study the…

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if $A\subset\mathbb{R}^2$ is a Borel set of Hausdorff dimension $s>1$, then its distance set has Hausdorff…

We are going to introduce a new algebraic, analytic structure that is a kind of generalization of the Hausdorff dimension and measure. We give many examples and study the basic properties and relations of such systems.

Let $\Theta=\{\theta_n\}, \Lambda=\{\lambda_n\}$ be two sequences of independent and identically distributed uniform random variables on $[0,1]$. The random vector-valued Weierstrass function is given by \[ f_{\Theta,\Lambda}(t)= \left(…

Let $\Lambda$ be a countable index set and $S=\{\phi_i: i\in \Lambda\}$ be a conformal iterated function system on $[0,1]^d$ satisfying the open set condition. Denote by $J$ the attractor of $S$. With each sequence $(w_1,w_2,...)\in…

We investigate the Hausdorff dimension of level sets defined by digit growth rates in $\theta$-expansions, a generalization of regular continued fractions. For any $\alpha \geq 0$, we prove that the set \[ E_\theta(\alpha) = \left\{ x \in…

This paper has been withdrawn Any real number $x$ in the unit interval can be expressed as a continued fraction $x=[n_1,...,n_{_N},...]$. Subsets of zero measure are obtained by imposing simple conditions on the $n_{_N}$. By imposing…

An approach is given for estimating the Hausdorff dimension of the univoque set of a self-similar set. This sometimes allows us to get the exact Hausdorff dimensions of the univoque sets.

We prove that, for every norm on $\mathbb{R}^d$ and every $E \subseteq \mathbb{R}^d$, the Hausdorff dimension of the distance set of $E$ with respect to that norm is at least $\dim_{\mathrm{H}} E - (d-1)$. An explicit construction follows,…

We prove that generically, for a self-affine set in $\mathbb{R}^d$, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension. This gives a partial positive answer to a folklore open…

Let $w=(w_1, w_2)$ be a pair of positive real numbers with $w_1+w_2=1$ and $w_1\ge w_2$. We show that the set of $w$-weighted singular vectors in $\mathbb R^2$ has Hausdorff dimension $2- \frac{1}{1+w_1}$. This extends the previous work of…

We consider infinite conformal iterated function systems on $\mathbb{R}^d$. We study the geometric structure of the limit set of such systems. Suppose this limit set intersects some $l$-dimensional $C^1$-submanifold with positive Hausdorff…

We show that any codimension one hyperbolic attractor of a diffeomorphism of a (d+1)-dimensional closed manifold is shape equivalent to a (d+1)-dimensional torus with a finite number of points removed, or, in the non-orientable case, to a…

We show that for each $d\in (0,2]$ there exists a meromorphic function $f$ such that the inverse function of $f$ has three singularities and the Julia set of $f$ has Hausdorff dimension $d$.

We study non-invertible piecewise hyperbolic maps in the plane. The Hausdorff dimension of the attractor is calculated in terms of the Lyapunov exponents, provided that the map satisfies a transversality condition. Explicit examples of maps…