Related papers: The $\beta$-transformation with a hole at 0

Let $T_{k}$ be the expanding map of $[0,1)$ defined by $T_{k}(x) = k x\ \text{mod 1}$, where $k\geq 2$ is an integer. Given $0\leq a<b\leq 1$, let $\mathcal{W}_{k}(a,b)=\{x\in [0,1)\ \vert \ T_{k}^nx\notin (a,b), \text{ for all } n\geq 0\}$…

We consider for each t the set K(t) of points of the circle whose forward orbit for the doubling map does not intersect (0,t), and look at the dimension function eta(t) := H.dim K(t). We prove that at every bifurcation parameter t, the…

We define a family B(t) of compact subsets of the unit interval which generalizes the sets of numbers whose continued fraction expansion has bounded digits. We study how the set B(t) changes as one moves the parameter t, and see that the…

We show that if $\beta>1$ is a rational number and the Julia set $J$ of the holomorphic correspondence $z^{\beta}+c$ is a locally eventually onto hyperbolic repeller, then the Hausdorff dimension of $J$ is bounded from above by the zero of…

Let $E$ be the self-similar set generated by the {\it iterated function system} {\[ f_0(x)=\frac{x}{\beta},\quad f_1(x)=\frac{x+1}{\beta}, \quad f_{\beta+1}=\frac{x+\beta+1}{\beta} \]}with $\beta\ge 3$. {Then} $E$ is a self-similar set with…

We consider the distribution of the orbits of the number 1 under the $\beta$-transformations $T_\beta$ as $\beta$ varies. Mainly, the size of the set of $\beta>1$ for which a given point can be well approximated by the orbit of 1 is…

For any $\beta > 1$, denoted by $r_n(x,\beta)$ the maximal length of consecutive zeros amongst the first $n$ digits of the $\beta$-expansion of $x\in[0,1]$. The limit superior (respectively limit inferior) of $\frac{r_n(x,\beta)}{n}$ is…

Given an integer $N\ge 2$ and a real number ${\beta}>1$, let $\Gamma_{{\beta},N}$ be the set of all $x=\sum_{i=1}^\infty {d_i}/{{\beta}^i}$ with $d_i\in\{0,1,\cdots,N-1\}$ for all $i\ge 1$. The infinite sequence $(d_i)$ is called a…

For $\alpha$ in $(0,1]$, a subset $E$ of $\RR$ is called Furstenberg set of type $\alpha$ or $F_\alpha$-set if for each direction $e$ in the unit circle there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff…

In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $\alpha,\beta\in(0,1]$, we will say that a set $E\subset \R^2$ is an $F_{\alpha\beta}$-set if…

Given an integer $b\ge 3$, let $T_b: [0,1)\to [0,1); x\mapsto bx\pmod 1$ be the expanding map on the unit circle. For any $m\in\mathbb{N}$ and $\omega=\omega^0\omega^1\ldots\in(\left\{0,1,\ldots,b-1\right\}^m)^\mathbb{N_0}$ let \[…

We construct a family $\{\Phi_t\}_{t\in[0,1]}$ of homeomorphisms of the two-torus isotopic to the identity, for which all of the rotation sets $\rho(\Phi_t)$ can be described explicitly. We analyze the bifurcations and typical behavior of…

By applying a 2014 result on the distribution of full cylinders, we give a proof of the useful folklore: for any $\beta>1$, the Hausdorff dimension of an arbitrary set in the shift space $S_\beta$ is equal to the Hausdorff dimension of its…

Given a positive integer $M$ and $q\in(1,M+1]$, let $\mathcal U_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion: there exists a unique sequence $(x_i)=x_1x_2\ldots$ with each $x_i\in\{0,1,\ldots, M\}$ such that \[…

In [Bak] the first author proved that for any $\beta\in (1,\beta_{KL})$ every $x\in(0,\frac{1}{\beta-1})$ has a simply normal $\beta$-expansion, where $\beta_{KL}\approx 1.78723$ is the Komornik-Loreti constant. This result is complemented…

We show that, for any $0<\gamma<1/2$, any $(\alpha,\beta)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrt{\gamma}$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such…

The purpose of this paper is to complete the proof of the following result. Let $0 < \beta \leq \alpha < 1$ and $\kappa > 0$. Then, there exists $\eta > 0$ such that whenever $A,B \subset \mathbb{R}$ are Borel sets with $\dim_{\mathrm{H}} A…

Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map satisfying a property we call almost specification (which is slightly weaker than the $g$-almost product property of Pfister and Sullivan), and $\phi$ be a…

Much has been written about expansions of real numbers in noninteger bases. Particularly, for a finite alphabet $\{0,1,\dots,\alpha\}$ and a real number (base) $1<\beta<\alpha+1$, the so-called {\em univoque set} of numbers which have a…

Let $\beta_1,\beta_2>1$ and $T_i(x,y) = \bigl(\frac{x+i}{\beta_1}, \frac{y+i}{\beta_2}\bigr),\ i\in\{\pm1\}$. Let $A := A_{\beta_1, \beta_2}$ be the unique compact set satisfying $A = T_{1}(A) \cup T_{-1}(A)$. In this paper we give a…