# The -transformation with a hole at 0

@article{Kalle2019TheW,
title={The -transformation with a hole at 0},
author={Charlene Kalle and Derong Kong and Niels Langeveld and Wenxia Li},
journal={Ergodic Theory and Dynamical Systems},
year={2019}
}
<jats:p>For <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385719000129_inline2.png" /><jats:tex-math>$\unicode[STIX]{x1D6FD}\in (1,2]$</jats:tex-math></jats:alternatives></jats:inline-formula> the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385719000129_inline3.png" /><jats…
4 Citations

## Figures from this paper

Two bifurcation sets arising from the beta transformation with a hole at 0
• Mathematics
• 2019
Given $\beta\in(1,2],$ the $\beta$-transformation $T_\beta: x\mapsto \beta x\pmod 1$ on the circle $[0, 1)$ with a hole $[0, t)$ was investigated by Kalle et al.~(2019). They described the set-valued
Pointwise densities of homogeneous Cantor measure and critical values
• Physics, Mathematics
• 2020
Let N ⩾ 2 and ρ ∈ (0, 1/N 2]. The homogenous Cantor set E is the self-similar set generated by the iterated function system fi(x)=ρx+i(1−ρ)N−1:i=0,1,…,N−1. Let s = dim H  E be the Hausdorff dimension
The bifurcation set as a topological invariant for one-dimensional dynamics
• Physics, Mathematics
• 2019
For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of (some
Generalized Fibonacci numbers and extreme value laws for the Rényi map
• Mathematics
• 2020
In this paper we prove an extreme value law for a stochastic process obtained by iterating the Renyi map. Haiman (2018) derived a recursion formula for the Lebesgue measure of threshold exceedance

## References

SHOWING 1-10 OF 62 REFERENCES
The $\beta$-transformation with a hole
This paper extends those of Glendinning and Sidorov [3] and of Hare and Sidorov [6] from the case of the doubling map to the more general $\beta$-transformation. Let $\beta \in (1,2)$ and consider
The doubling map with asymmetrical holes
• Mathematics
Ergodic Theory and Dynamical Systems
• 2013
Abstract Let $0\lt a\lt b\lt 1$ and let $T$ be the doubling map. Set $\mathcal{J} (a, b): = \{ x\in [0, 1] : {T}^{n} x\not\in (a, b), n\geq 0\}$. In this paper we completely characterize the holes
A NOTE ON INHOMOGENEOUS DIOPHANTINE APPROXIMATION IN BETA-DYNAMICAL SYSTEM
• Mathematics
Bulletin of the Australian Mathematical Society
• 2014
Abstract We study the distribution of the orbits of real numbers under the beta-transformation $T_{{\it\beta}}$ for any ${\it\beta}>1$. More precisely, for any real number ${\it\beta}>1$ and a
Relative bifurcation sets and the local dimension of univoque bases
• Mathematics
Ergodic Theory and Dynamical Systems
• 2021
Abstract Fix an alphabet $A=\{0,1,\ldots ,M\}$ with $M\in \mathbb{N}$ . The univoque set $\mathscr{U}$ of bases $q\in (1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has
Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions
• Mathematics
• 2013
We consider the distribution of the orbits of the number 1 under the $$\beta$$β-transformations $$T_\beta$$Tβ as $$\beta$$β varies. Mainly, the size of the set of $$\beta >1$$β>1 for which a given
Supercritical Holes for the Doubling Map
The purpose of this note is to completely characterize all supercritical holes for the doubling map Tx = 2x mod 1.
A two-dimensional univoque set
• Mathematics
• 2010
Let $\mathbf{J} \subset \mathbb{R}^2$ be the set of couples $(x,q)$ with $q>1$ such that $x$ has at least one representation of the form $x=\sum_{i=1}^{\infty} c_i q^{-i}$ with integer coefficients
An algebraic approach to entropy plateaus in non-integer base expansions
For a positive integer $M$ and a real base $q\in(1,M+1]$, let $\mathcal{U}_q$ denote the set of numbers having a unique expansion in base $q$ over the alphabet $\{0,1,\dots,M\}$, and let
Critical base for the unique codings of fat Sierpinski gasket
• Mathematics
• 2018
Given $\beta\in(1,2)$ the fat Sierpinski gasket $\mathcal S_\beta$ is the self-similar set in $\mathbb R^2$ generated by the iterated function system (IFS) \[ f_{\beta,d}(x)=\frac{x+d}{\beta},\quad
Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions
• Mathematics
• 2014
Letˇ > 1 be a real number. LetTˇ denote theˇ-transformation on Œ0; 1 . A cylinder of order n is a set of real numbers in Œ0; 1 having the same first n digits in their ˇ-expansion. A cylinder is