Zobrazeno 1 - 10
of 162
pro vyhledávání: '"Allaart, Pieter"'
Autor:
Allaart, Pieter, Kong, Derong
Given $\beta>1$, let $T_\beta$ be the $\beta$-transformation on the unit circle $[0,1)$, defined by $T_\beta(x)=\beta x\pmod 1$. For each $t\in[0,1)$ let $K_\beta(t)$ be the survivor set consisting of all $x\in[0,1)$ whose orbit $\{T^n_\beta(x): n\ge
Externí odkaz:
http://arxiv.org/abs/2411.03516
Autor:
Allaart, Pieter, Jones, Taylor
We introduce a natural way to construct a random subset of a homogeneous Cantor set $C$ in $[0,1]$ via random labelings of an infinite $M$-ary tree, where $M\geq 2$. The Cantor set $C$ is the attractor of an equicontractive iterated function system $
Externí odkaz:
http://arxiv.org/abs/2308.04569
Autor:
Allaart, Pieter, Kong, Derong
Given $\beta\in(1,2]$, let $T_\beta$ be the $\beta$-transformation on the unit circle $[0,1)$ such that $T_\beta(x)=\beta x\pmod 1$. For each $t\in[0,1)$ let $K_\beta(t)$ be the survivor set consisting of all $x\in[0,1)$ whose orbit $\{T^n_\beta(x):
Externí odkaz:
http://arxiv.org/abs/2304.06892
Autor:
Allaart, Pieter, Jones, Taylor
Publikováno v:
J. Math. Anal. Appl. 521 (2022), no. 2, Article 126909
We consider a class of "box-like" statistically self-affine functions, and compute the almost-sure box-counting dimension of their graphs. Furthermore, we consider the differentiability of our functions, and prove that, depending on an explicitly com
Externí odkaz:
http://arxiv.org/abs/2208.00035
Autor:
Allaart, Pieter, Kong, Derong
Given $\beta\in(1,2]$, let $T_\beta$ be the $\beta$-transformation on the unit circle $[0,1)$ such that $T_\beta(x)=\beta x\pmod 1$. For each $t\in[0,1)$ let $K_\beta(t)$ be the survivor set consisting of all $x\in[0,1)$ whose orbit $\{T^n_\beta(x):
Externí odkaz:
http://arxiv.org/abs/2109.10012
A Trott number is a number $x\in(0,1)$ whose continued fraction expansion is equal to its base $b$ expansion for a given base $b$, in the following sense: If $x=[0;a_1,a_2,\dots]$, then $x=(0.\hat{a}_1\hat{a}_2\dots)_b$, where $\hat{a}_i$ is the stri
Externí odkaz:
http://arxiv.org/abs/2108.03664
Autor:
Allaart, Pieter, Kong, Derong
Publikováno v:
Nonlinearity 35 (2022), 6453-6484
Given $\rho\in(0, 1/3]$, let $\mu$ be the Cantor measure satisfying $\mu=\frac{1}{2}\mu f_0^{-1}+\frac{1}{2}\mu f_1^{-1}$, where $f_i(x)=\rho x+i(1-\rho)$ for $i=0, 1$. The support of $\mu$ is a Cantor set $C$ generated by the iterated function syste
Externí odkaz:
http://arxiv.org/abs/2008.04474
Autor:
Allaart, Pieter, Kong, Derong
Publikováno v:
Trans. Amer. Math. Soc. 374 (2021), no. 9, 6201-6249
Given a real number $x>0$, we determine $q_s(x):=\inf\mathscr{U}(x)$, where $\mathscr{U}(x)$ is the set of all bases $q\in(1,2]$ for which $x$ has a unique expansion of $0$'s and $1$'s. We give an explicit description of $q_s(x)$ for several regions
Externí odkaz:
http://arxiv.org/abs/2006.07927
Autor:
Allaart, Pieter
Publikováno v:
J. Math. Anal. Appl. 488, Article 124096 (2020)
This paper gives the pointwise H\"older (or multifractal) spectrum of continuous functions on the interval $[0,1]$ whose graph is the attractor of an iterated function system consisting of $r\geq 2$ affine maps on $\mathbb{R}^2$. These functions sati
Externí odkaz:
http://arxiv.org/abs/1907.09660
Autor:
Allaart, Pieter C.
Publikováno v:
Discrete Contin. Dyn. Syst. A 39, no. 11, 6507--6522 (2019)
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 $\mathbf{U}_q$ denote the corresponding set of sequences in $\{
Externí odkaz:
http://arxiv.org/abs/1812.09446