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Given a self-similar iterated function system $\Phi=\{ \phi_i(x)=\rho_i O_i x+t_i \}_{i=1}^m$ acting on $\mathbb{R}^d$, we can generate a parameterised family of iterated function systems by replacing each $t_i$ with a random vector in $\mathbb{R}^d$. In this paper we study whether a Lebesgue typical member of this family will satisfy the strong separation condition. Our main results show that if the similarity dimension of $\Phi$ is sufficiently small, then a Lebesgue typical member of this family will satisfy the strong separation condition., Comment: The main results are covered by Theorem 1.1 in [M. Rams and J. L. V\'ehel, Results on the dimension spectrum for self-conformal measures. Nonlinearity 20 (2007), no.4, 965-973. https://iopscience.iop.org/article/10.1088/0951-7715/20/4/009]

For $\beta>1$ let $S_\beta$ be the Sierpinski gasket generated by the iterated function system \[\left\{f_{\alpha_0}(x,y)=\Big(\frac{x}{\beta},\frac{y}{\beta}\Big), \quad f_{\alpha_1}(x,y)=\Big(\frac{x+1}{\beta}, \frac{y}{\beta}\Big), \quad f_{\alpha_2}(x,y)=\Big(\frac{x}{\beta}, \frac{y+1}{\beta}\Big)\right\}.\] If $\beta\in(1,2]$, then the overlap region $O_\beta:=\bigcup_{i\ne j}f_{\alpha_i}(\Delta_\beta)\cap f_{\alpha_j}(\Delta_\beta)$ is nonempty, where $\Delta_\beta$ is the convex hull of $S_\beta$. In this paper we study the periodic codings of the univoque set \[ \mathbf U_\beta:=\left\{(d_i)_{i=1}^\infty\in\{(0,0), (1,0), (0,1)\}^\mathbb N: \sum_{i=1}^\infty d_{n+i}\beta^{-i}\in S_\beta\setminus O_\beta~\forall n\ge 0\right\}. \] More precisely, we determine for each $k\in\mathbb N$ the smallest base $\beta_k\in(1,2]$ such that for any $\beta>\beta_k$ the set $\mathbf U_\beta$ contains a sequence of smallest period $k$. We show that each $\beta_k$ is a Perron number, and the sequence $(\beta_k)$ has infinitely many accumulation points. Furthermore, we show that $\beta_{3k}>\beta_{3\ell}$ if and only if $k$ is larger than $\ell$ in the Sharkovskii ordering; and the sequences $ (\beta_{3\ell+1}), (\beta_{3\ell+2})$ decreasingly converge to the same limit point $\beta_a\approx 1.55898$, respectively. In particular, we find that $\beta_{6m+4}=\beta_{3m+2}$ for all $m\ge 0$. Consequently, we prove that if $\mathbf U_\beta$ contains a sequence of smallest period $2$ or $4$, then $\mathbf U_\beta$ contains a sequence of smallest period $k$ for any $k\in\mathbb N$., Comment: 34 pages, 6 figures

In this paper we introduce two notions of fractal sumset properties. A compact set $K\subset\mathbb{R}^d$ is said to have the Hausdorff sumset property (HSP) if for any $\ell\in\mathbb{N}_{\ge 2}$ there exist compact sets $K_1, K_2,\ldots, K_\ell$ such that $K_1+K_2+\cdots+K_\ell\subset K$ and $\dim_H K_i=\dim_H K$ for all $1\le i\le \ell$. Analogously, if we replace the Hausdorff dimension by the packing dimension in the definition of HSP, then the compact set $K\subset\mathbb{R}^d$ is said to have the packing sumset property (PSP). We show that the HSP fails for certain homogeneous self-similar sets satisfying the strong separation condition, while the PSP holds for all homogeneous self-similar sets in $\mathbb{R}^d$., Comment: 11 pages, final version

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 $\{T^n_\beta(x): n\ge 0\}$ never enters the interval $[0,t)$. Letting $\mathscr{E}_\beta$ denote the bifurcation set of the set-valued map $t\mapsto K_\beta(t)$, Kalle et al. [Ergodic Theory Dynam. Systems, 40 (9): 2482--2514, 2020] conjectured that \[ \dim_H\big(\mathscr{E}_\beta\cap[t,1]\big)=\dim_H K_\beta(t) \qquad \forall\,t\in(0,1). \] The main purpose of this article is to prove this conjecture. We do so by investigating dynamical properties of the symbolic equivalent of the survivor set $K_\beta(t)$, in particular its entropy and topological transitivity. In addition, we compare $\mathscr{E}_\beta$ with the bifurcation set $\mathscr{B}_\beta$ of the map $t\mapsto \dim_H K_\beta(t)$ (which is a decreasing devil's staircase by a theorem of Kalle et al.), and show that, for Lebesgue-almost every $\beta\in(1,2]$, the difference $\mathscr{E}_\beta\backslash\mathscr{B}_\beta$ has positive Hausdorff dimension, but for every $k\in\{0,1,2,\dots\}\cup\{\aleph_0\}$, there are infinitely many values of $\beta$ such that the cardinality of $\mathscr{E}_\beta\backslash\mathscr{B}_\beta$ is exactly $k$. For a countable but dense subset of $\beta$'s, we also determine the intervals of constancy of the function $t\mapsto \dim_H K_\beta(t)$. Some connections with other topics in dynamics, such as kneading invariants of Lorenz maps and the doubling map with an arbitrary hole, are also discussed., Comment: 66 pages, 1 figure. One more theorem was added (Theorem 1.4) and the paper was thoroughly reorganized

Fix a positive integer $N$ and a real number $0< \beta < 1/(N+1)$. Let $\Gamma$ be the homogeneous symmetric Cantor set generated by the IFS $$ \Big\{ \phi_i(x)=\beta x + i \frac{1-\beta}{N}: i=0,1,\cdots, N \Big\}. $$ For $m\in\mathbb{Z}_+$ we show that there exist infinitely many translation vectors $\mathbf t=(t_0,t_1,\cdots, t_m)$ with $0=t_0

Given $\rho\in (0,1/4]$, the four corner Cantor set $E\subset \mathbb{R}^{2}$ is a self-similar set generated by the iterated function system \[ \left\{(\rho x, \rho y), \quad(\rho x, \rho y+1-\rho),\quad (\rho x+1-\rho, \rho y),\quad(\rho x+1-\rho,\rho y+1-\rho)\right\}. \] For $\theta\in[0,\pi)$ let $E_\theta$ be the orthogonal projection of $E$ onto a line with an angle $\theta$ to the $x$-axis. In this paper we give a complete characterization on which the projection $E_\theta$ is totally self-similar. We also study the spectrum of $E_\theta $, which turns out that the spectrum of $E_\theta$ achieves its maximum value if and only if $E_\theta $ is totally self-similar. Furthermore, when $E_\theta$ is totally self-similar, we calculate its Hausdorff dimension and study the subset $U_\theta $ which consists of all $x\in E_\theta $ having a unique coding. In particular, we show that $\dim_H U_\theta=\dim_H E_\theta$ for Lebesgue almost every $\theta \in[0,\pi)$. Finally, for $\rho=1/4$ we describe the distribution of $\theta $ in which $E_\theta$ contains an interval. It turns out that the possibility for $E_\theta$ to contain an interval is smaller than that for $E_\theta$ to have an exact overlap., Comment: 33 pages, 3 figures

Given a positive integer $M$ and a real number $x\in(0,1]$, we call $q\in(1,M+1]$ a univoque simply normal base of $x$ if there exists a unique simply normal sequence $(d_i)\in\{0,1,\ldots,M\}^\mathbb N$ such that $x=\sum_{i=1}^\infty d_i q^{-i}$. Similarly, a base $q\in(1,M+1]$ is called a univoque irregular base of $x$ if there exists a unique sequence $(d_i)\in\{0,1,\ldots, M\}^\mathbb N$ such that $x=\sum_{i=1}^\infty d_i q^{-i}$ and the sequence $(d_i)$ has no digit frequency. Let $\mathcal U_{SN}(x)$ and $\mathcal U_{I_r}(x)$ be the sets of univoque simply normal bases and univoque irregular bases of $x$, respectively. In this paper we show that for any $x\in(0,1]$ both $\mathcal U_{SN}(x)$ and $\mathcal U_{I_r}(x)$ have full Hausdorff dimension. Furthermore, given finitely many rationals $x_1, x_2, \ldots, x_n\in(0,1]$ so that each $x_i$ has a finite expansion in base $M+1$, we show that there exists a full Hausdorff dimensional set of $q\in(1,M+1]$ such that each $x_i$ has a unique expansion in base $q$., Comment: 25 pages, 2 figures

Given two coprime integers $p\ge 2$ and $q \ge 3$, let $D_p\subset[0,1)$ consist of all rational numbers which have a finite $p$-ary expansion, and let $$ K(q, \mathcal{A})=\bigg\{ \sum_{i=1}^\infty \frac{d_i}{q^i}: d_i\in \mathcal{A}~ \forall i\in\mathbb{N} \bigg\}, $$ where $\mathcal{A} \subset \{0,1,\ldots, q-1\}$ with cardinality $1<\#\mathcal{A}< q$. In 2021 Schleischitz showed that $\#(D_p\cap K(q,\mathcal{A}))<+\infty$. In this paper we show that for any $r\in\mathbb{Q}$ and for any $\alpha\in\mathbb{R}$, $$ \#\big((r D_p+\alpha)\cap K(q,\mathcal{A})\big)<+\infty. $$, Comment: 8 pages, final version

For $\lambda\in(0,1/3]$ let $C_\lambda$ be the middle-$(1-2\lambda)$ Cantor set in $\mathbb R$. Given $t\in[-1,1]$, excluding the trivial case we show that \[ \Lambda(t):=\left\{\lambda\in(0,1/3]: C_\lambda\cap(C_\lambda+t)\ne\emptyset\right\} \] is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of $\Lambda(t)$, which reveals a dimensional variation principle. Furthermore, for any $\beta\in[0,1]$ we show that the level set \[ \Lambda_\beta(t):=\left\{\lambda\in\Lambda(t): \dim_H(C_\lambda\cap(C_\lambda+t))=\dim_P(C_\lambda\cap(C_\lambda+t))=\beta\frac{\log 2}{-\log \lambda}\right\} \] has equal Hausdorff and packing dimension $(-\beta\log\beta-(1-\beta)\log\frac{1-\beta}{2})/\log 3$. We also show that the set of $\lambda\in\Lambda(t)$ for which $\dim_H(C_\lambda\cap(C_\lambda+t))\ne\dim_P(C_\lambda\cap(C_\lambda+t))$ has full Hausdorff dimension., Comment: 32 pages, 3 figures

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 $\{T^n_\beta(x): n\ge 0\}$ never hits the open interval $(0,t)$. Kalle et al. proved in [Ergodic Theory Dynam. Systems, 40 (9): 2482--2514, 2020] that the Hausdorff dimension function $t\mapsto\dim_H K_\beta(t)$ is a non-increasing Devil's staircase. So there exists a critical value $\tau(\beta)$ such that $\dim_H K_\beta(t)>0$ if and only if $t<\tau(\beta)$. In this paper we determine the critical value $\tau(\beta)$ for all $\beta\in(1,2]$, answering a question of Kalle et al. (2020). For example, we find that for the Komornik-Loreti constant $\beta\approx 1.78723$ we have $\tau(\beta)=(2-\beta)/(\beta-1)$. Furthermore, we show that (i) the function $\tau: \beta\mapsto\tau(\beta)$ is left continuous on $(1,2]$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $\tau$ has no downward jumps, with $\tau(1+)=0$ and $\tau(2)=1/2$; and (iii) there exists an open set $O\subset(1,2]$, whose complement $(1,2]\setminus O$ has zero Hausdorff dimension, such that $\tau$ is real-analytic, convex and strictly decreasing on each connected component of $O$. Consequently, the dimension $\dim_H K_\beta(t)$ is not jointly continuous in $\beta$ and $t$. Our strategy to find the critical value $\tau(\beta)$ depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems., Comment: 40 pages, 2 figures and 1 table

For $\lambda\in(0,1/2]$ let $K_\lambda \subset\mathbb{R}$ be a self-similar set generated by the iterated function system $\{\lambda x, \lambda x+1-\lambda\}$. Given $x\in(0,1/2)$, let $\Lambda(x)$ be the set of $\lambda\in(0,1/2]$ such that $x\in K_\lambda$. In this paper we show that $\Lambda(x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\ldots, y_p\in(0,1/2)$ there exists a full Hausdorff dimensional set of $\lambda\in(0,1/2]$ such that $y_1,\ldots, y_p \in K_\lambda$., Comment: 21 pages, 1 figure, final version

Let $m\in\mathbb N_{\ge 2}$, and let $\mathcal K=\{K_\lambda: \lambda\in(0, 1/m]\}$ be a class of Cantor sets, where $K_{\lambda}=\{\sum_{i=1}^\infty d_i\lambda^i: d_i\in\{0,1,\ldots, m-1\}, i\ge 1\}$. We investigate in this paper the likelyhood of a fixed point in the Cantor sets of $\mathcal K$. More precisely, for a fixed point $x\in(0,1)$ we consider the parameter set $\Lambda(x)=\{\lambda\in(0,1/m]: x\in K_\lambda\}$, and show that $\Lambda(x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, by constructing a sequence of Cantor subsets with large thickness in $\Lambda(x)$ we prove that the intersection $\Lambda(x)\cap\Lambda(y)$ also has full Hausdorff dimension for any $x, y\in(0,1)$., Comment: 29 pages, 5 figures. We added some statements in the abstract and the introduction

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 system $\{f_0, f_1\}$. Continuing the work of Feng et al. (2000) on the pointwise lower and upper densities \[ \Theta_*^s(\mu, x)=\liminf_{r\to 0}\frac{\mu(B(x,r))}{(2r)^s},\qquad \Theta^{*s}(\mu, x)=\limsup_{r\to 0}\frac{\mu(B(x,r))}{(2r)^s}, \] where $s=-\log 2/\log\rho$ is the Hausdorff dimension of $C$, we give a complete description of the sets $D_*$ and $D^*$ consisting of all possible values of the lower and upper densities, respectively. We show that both sets contain infinitely many isolated and infinitely many accumulation points, and they have the same Hausdorff dimension as the Cantor set $C$. Furthermore, we compute the Hausdorff dimension of the level sets of the lower and upper densities. Our method consists in formulating an equivalent ``dyadic" version of the problem involving the doubling map on $[0,1)$, which we solve by using known results on the entropy of a certain open dynamical system and the notion of tuning., Comment: 30 pages, 2 figures

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 of $x$-values. For others, we present an efficient algorithm to determine $q_s(x)$ and the lexicographically smallest unique expansion of $x$. We show that the infimum is attained for almost all $x$, but there is also a set of points of positive Hausdorff dimension for which the infimum is proper. In addition, we show that the function $q_s$ is right-continuous with left-hand limits and no downward jumps, and characterize the points of discontinuity of $q_s$. A large part of the paper is devoted to the level sets $L(q):=\{x>0:q_s(x)=q\}$. We show that $L(q)$ is finite for almost every $q$, but there are also infinitely many infinite level sets. In particular, for the Komornik-Loreti constant $q_{KL}=\min\mathscr{U}(1)\approx 1.787$ we prove that $L(q_{KL})$ has both infinitely many left- and infinitely many right accumulation points., Comment: 50 pages, 2 figures. Added section 8 on the maximum value of q_s(x) and fixed some typos

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 complete {overlaps}, i.e., $f_{0}\circ f_{\beta+1}=f_{1}\circ f_1$, but $E$ is not totally self-similar. We investigate all its generating iterated function systems, give the spectrum of $E$, and determine the Hausdorff dimension and Hausdorff measure of $E$ and of the sets which contain all points in $E$ having finite or infinite different triadic codings., Comment: 17 pages, 1 figure

Let $N\ge 2$ and $\rho\in(0,1/N^2]$. The homogenous Cantor set $E$ is the self-similar set generated by the iterated function system \[ \left\{f_i(x)=\rho x+\frac{i(1-\rho)}{N-1}: i=0,1,\ldots, N-1\right\}. \] Let $s=\dim_H E$ be the Hausdorff dimension of $E$, and let $\mu=\mathcal H^s|_E$ be the $s$-dimensional Hausdorff measure restricted to $E$. In this paper we describe, for each $x\in E$, the pointwise lower $s$-density $\Theta_*^s(\mu,x)$ and upper $s$-density $\Theta^{*s}(\mu, x)$ of $\mu$ at $x$. This extends some early results of Feng et al. (2000). Furthermore, we determine two critical values $a_c$ and $b_c$ for the sets \[ E_*(a)=\left\{x\in E: \Theta_*^s(\mu, x)\ge a\right\}\quad\textrm{and}\quad E^*(b)=\left\{x\in E: \Theta^{*s}(\mu, x)\le b\right\} \] respectively, such that $\dim_H E_*(a)>0$ if and only if $a0$ if and only if $b>b_c$. We emphasize that both values $a_c$ and $b_c$ are related to the Thue-Morse type sequences, and our strategy to find them relies on ideas from open dynamics and techniques from combinatorics on words., Comment: 30 pages, 1 figure and 1 table

Given a positive integer $M$ and a real number $x>0$, let $\mathcal U(x)$ be the set of all bases $q\in(1, M+1]$ for which there exists a unique sequence $(d_i)=d_1d_2\ldots$ with each digit $d_i\in\{0,1,\ldots, M\}$ satisfying $$ x=\sum_{i=1}^\infty\frac{d_i}{q^i}. $$ The sequence $(d_i)$ is called a $q$-expansion of $x$. In this paper we investigate the local dimension of $\mathcal U(x)$ and prove a `variation principle' for unique non-integer base expansions. We also determine the critical values of $\mathcal U(x)$ such that when $x$ passes the first critical value the set $\mathcal U(x)$ changes from a set with positive Hausdorff dimension to a countable set, and when $x$ passes the second critical value the set $\mathcal U(x)$ changes from an infinite set to a singleton. Denote by $\mathbf U(x)$ the set of all unique $q$-expansions of $x$ for $q\in\mathcal U(x)$. We give the Hausdorff dimension of $\mathbf U(x)$ and show that the dimensional function $x\mapsto\dim_H\mathbf U(x)$ is a non-increasing Devil's staircase. Finally, we investigate the topological structure of $\mathcal U(x)$. In contrast with $x=1$ that $\mathcal U(1)$ has no isolated points, we prove that for typical $x>0$ the set $\mathcal U(x)$ contains isolated points., Comment: 26 pages, 2 figures. In this version we simplified the proof of Theorem 1.1

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 bifurcation set \[ \mathcal E_\beta:=\{t\in[0, 1): K_\beta(t')\ne K_\beta(t)~\forall t'>t\}, \] where $K_\beta(t):=\{x\in[0, 1): T_\beta^n(x)\ge t~\forall n\ge 0\}$ is the survivor set. In this paper we investigate the dimension bifurcation set \[ \mathcal B_\beta:=\{t\in[0, 1): \dim_H K_\beta(t')\ne \dim_H K_\beta(t)~\forall t'>t\}, \] where $\dim_H$ denotes the Hausdorff dimension. We show that if $\beta\in(1,2]$ is a multinacci number then the two bifurcation sets $\mathcal B_\beta$ and $\mathcal E_\beta$ coincide. Moreover we give a complete characterization of these two sets. As a corollary of our main result we prove that for $\beta$ a multinacci number we have $\dim_H(\mathcal E_\beta\cap[t, 1])=\dim_H K_\beta(t)$ for any $t\in[0, 1)$. This confirms a conjecture of Kalle et al.~for $\beta$ a multinacci number., Comment: 12 pages

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) \[ f_{\beta,d}(x)=\frac{x+d}{\beta},\quad d\in\mathcal A:=\{(0, 0), (1,0), (0,1)\}. \] Then for each point $P\in\mathcal S_\beta$ there exists a sequence $(d_i)\in\mathcal A^\mathbb N$ such that $P=\sum_{i=1}^\infty d_i/\beta^i$, and the infinite sequence $(d_i)$ is called a \emph{coding} of $P$. In general, a point in $\mathcal S_\beta$ may have multiple codings since the overlap region $\mathcal O_\beta:=\bigcup_{c,d\in\mathcal A, c\ne d}f_{\beta,c}(\Delta_\beta)\cap f_{\beta,d}(\Delta_\beta)$ has non-empty interior, where $\Delta_\beta$ is the convex hull of $\mathcal S_\beta$. In this paper we are interested in the invariant set \[ \widetilde{\mathcal U}_\beta:=\left\{\sum_{i=1}^\infty \frac{d_i}{\beta^i}\in \mathcal S_\beta: \sum_{i=1}^\infty\frac{d_{n+i}}{\beta^i}\notin\mathcal O_\beta~\forall n\ge 0\right\}. \] Then each point in $ \widetilde{\mathcal U}_\beta$ has a unique coding. We show that there is a transcendental number $\beta_c\approx 1.55263$ related to the Thue-Morse sequence, such that $\widetilde{\mathcal U}_\beta$ has positive Hausdorff dimension if and only if $\beta>\beta_{c}$. Furthermore, for $\beta=\beta_c$ the set $\widetilde{\mathcal U}_\beta$ is uncountable but has zero Hausdorff dimension, and for $\beta<\beta_c$ the set $\widetilde{\mathcal U}_\beta$ is at most countable. Consequently, we also answer a conjecture of Sidorov (2007). Our strategy is using combinatorics on words based on the lexicographical characterization of $\widetilde{\mathcal U}_\beta$., Comment: 28 pages, 10 figures

Given $\lambda\in(0, 1)$, let $E_\lambda$ be the self-similar set generated by the iterated function system $\{x/3,(x+\lambda)/3,(x+2)/3\}$. Then $E_\lambda$ is a self-similar set with overlaps. We obtain the necessary and sufficient condition for $E_\lambda$ to be totally self-similar, which is a concept first introduced by Broomhead, Montaldi, and Sidorov in 2004. When $E_\lambda$ is totally self-similar, all its generating IFSs are investigated, and the size of the set of points having finite triadic codings is determined. Besides, we give some properties of the spectrum of $E_\lambda$ and show that the spectrum of $E_\lambda$ vanishes if and only if $\lambda$ is irrational., Comment: 28 pages, 1 figure

Let $F=\{\mathbf{p}_0,\ldots,\mathbf{p}_n\}$ be a collection of points in $\mathbb{R}^d.$ The set $F$ naturally gives rise to a family of iterated function systems consisting of contractions of the form $$S_i(\mathbf{x})=\lambda \mathbf{x} +(1-\lambda)\mathbf{p}_i,$$ where $\lambda \in(0,1)$. Given $F$ and $\lambda$ it is well known that there exists a unique non-empty compact set $X$ satisfying $X=\cup_{i=0}^n S_i(X)$. For each $\mathbf{x} \in X$ there exists a sequence $\mathbf{a}\in\{0,\ldots,n\}^{\mathbb{N}}$ satisfying $$\mathbf{x}=\lim_{j\to\infty}(S_{a_1}\circ \cdots \circ S_{a_j})(\mathbf{0}).$$ We call such a sequence a coding of $\mathbf{x}$. In this paper we prove that for any $F$ and $k \in\mathbb{N},$ there exists $\delta_k(F)>0$ such that if $\lambda\in(1-\delta_k(F),1),$ then every point in the interior of $X$ has a coding which is $k$-simply normal. Similarly, we prove that there exists $\delta_{uni}(F)>0$ such that if $\lambda\in(1-\delta_{uni}(F),1),$ then every point in the interior of $X$ has a coding containing all finite words. For some specific choices of $F$ we obtain lower bounds for $\delta_k(F)$ and $\delta_{uni}(F)$. We also prove some weaker statements that hold in the more general setting when the similarities in our iterated function systems exhibit different rates of contraction. Our proofs rely on a variation of a well known construction of a normal number due to Champernowne, and an approach introduced by Erd\H{o}s and Komornik.

Fix an alphabet $A=\{0,1,\dots,M\}$ 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 been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper investigates how the set $\mathscr{U}$ is distributed over the interval $(1,M+1)$ by determining the limit $$f(q):=\lim_{\delta\to 0}\dim_H\big(\mathscr{U}\cap(q-\delta,q+\delta)\big)$$ for all $q\in(1,M+1)$. We show in particular that $f(q)>0$ if and only if $q\in\overline{\mathscr{U}}\backslash\mathscr{C}$, where $\mathscr{C}$ is an uncountable set of Hausdorff dimension zero, and $f$ is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of $\mathscr{U}$ called {\emph relative bifurcation sets}, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval, answering a question of Kalle et al.~[{\emph arXiv:1612.07982; to appear in Acta Arithmetica}, 2018]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [{\emph Adv. Math.}, 308:575--598, 2017] about strongly univoque sets., Comment: 31 pages and 1 figure

In a recent paper [Adv. Math. 305:165--196, 2017], Komornik et al.~proved a long-conjectured formula for the Hausdorff dimension of the set $\mathcal{U}_q$ of numbers having a unique expansion in the (non-integer) base $q$, and showed that this Hausdorff dimension is continuous in $q$. Unfortunately, their proof contained a gap which appears difficult to fix. This article gives a completely different proof of these results, using a more direct combinatorial approach., Comment: 18 pages

For $\beta\in(1,2]$ the $\beta$-transformation $T_\beta: [0,1) \to [0,1)$ is defined by $T_\beta ( x) = \beta x \pmod 1$. For $t\in[0, 1)$ let $K_\beta(t)$ be the survivor set of $T_\beta$ with hole $(0,t)$ given by \[K_\beta(t):=\{x\in[0, 1): T_\beta^n(x)\not \in (0, t) \textrm{ for all }n\ge 0\}.\] In this paper we characterise the bifurcation set $E_\beta$ of all parameters $t\in[0,1)$ for which the set valued function $t\mapsto K_\beta(t)$ is not locally constant. We show that $E_\beta$ is a Lebesgue null set of full Hausdorff dimension for all $\beta\in(1,2)$. We prove that for Lebesgue almost every $\beta\in(1,2)$ the bifurcation set $E_\beta$ contains both infinitely many isolated and accumulation points arbitrarily close to zero. On the other hand, we show that the set of $\beta\in(1,2)$ for which $E_\beta$ contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for $E_2$, the bifurcation set of the doubling map. Finally, we give for each $\beta \in (1,2)$ a lower and upper bound for the value $\tau_\beta$, such that the Hausdorff dimension of $K_\beta(t)$ is positive if and only if $t< \tau_\beta$. We show that $\tau_\beta \le 1-\frac1{\beta}$ for all $\beta \in (1,2)$., Comment: 32 pages, 4 figures

Given $x\in(0, 1]$, let $\mathcal U(x)$ be the set of bases $q\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^\infty d_i/q^i$. L\"{u}, Tan and Wu (2014) proved that $\mathcal U(x)$ is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum $\mathcal U(x)+\lambda\mathcal U(x)$ and product $\mathcal U(x)\cdot\mathcal U(x)^\lambda$ contain an interval for all $x\in(0, 1]$ and $\lambda\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (2017)., Comment: 21 pages, 1 figure. To appear in Indag. Math

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 by an observation made in [JSS], where it was shown that whenever $\beta\in (\beta_T, 2]$ there exists an $x\in(0,\frac{1}{\beta-1})$ with a unique $\beta$-expansion, and this expansion is not simply normal. Here $\beta_T\approx 1.80194$ is the unique zero in $(1,2]$ of the polynomial $x^3-x^2-2x+1$. This leaves a gap in our understanding within the interval $[\beta_{KL}, \beta_T]$. In this paper we fill this gap and prove that for any $\beta\in (1,\beta_T],$ every $x\in(0,\frac{1}{\beta-1})$ has a simply normal $\beta$-expansion. For completion, we provide a proof that for any $\beta\in(1,2)$, Lebesgue almost every $x$ has a simply normal $\beta$-expansion. We also give examples of $x$ with multiple $\beta$-expansions, none of which are simply normal. Our proofs rely on ideas from combinatorics on words and dynamical systems., Comment: 28 pages, 6 figures

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 \[ x=\frac{x_1}{q}+\frac{x_2}{q^2}+\frac{x_3}{q^3}+\cdots. \] Denote by $\mathbf U_q$ the set of corresponding sequences of all points in $\mathcal U_q$. It is well-known that the function $H: q\mapsto h(\mathbf U_q)$ is a Devil's staircase, where $h(\mathbf U_q)$ denotes the topological entropy of $\mathbf U_q$. In this paper we {give several characterizations of} the bifurcation set \[ \mathcal B:=\{q\in(1,M+1]: H(p)\ne H(q)\textrm{ for any }p\ne q\}. \] Note that $\mathcal B$ is contained in the set $\mathcal{U}^R$ of bases $q\in(1,M+1]$ such that $1\in\mathcal U_q$. By using a transversality technique we also calculate the Hausdorff dimension of the difference $\mathcal B\backslash\mathcal{U}^R$. Interestingly this quantity is always strictly between $0$ and $1$. When $M=1$ the Hausdorff dimension of $\mathcal B\backslash\mathcal{U}^R$ is $\frac{\log 2}{3\log \lambda^*}\approx 0.368699$, where $\lambda^*$ is the unique root in $(1, 2)$ of the equation $x^5-x^4-x^3-2x^2+x+1=0$., Comment: 28 pages, 1 figures and 1 table. To appear in J. Fractal Geometry

In this paper we answer several questions raised by Sidorov on the set $\mathcal B_2$ of bases in which there exist numbers with exactly two expansions. In particular, we prove that the set $\mathcal B_2$ is closed, and it contains both infinitely many isolated and accumulation points in $(1, q_{KL})$, where $q_{KL}\approx 1.78723$ is the Komornik-Loreti constant. Consequently we show that the second smallest element of $\mathcal B_2$ is the smallest accumulation point of $\mathcal B_2$. We also investigate the higher order derived sets of $\mathcal B_2$. Finally, we prove that there exists a $\delta>0$ such that \begin{equation*} \dim_H(\mathcal B_2\cap(q_{KL}, q_{KL}+\delta))<1, \end{equation*} where $\dim_H$ denotes the Hausdorff dimension., Comment: 34 pages, 1 figure. To appear in J. Number Theory

Given a positive integer $M$, for $q\in(1, M+1]$ let ${\mathcal{U}}_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion with the digit set $\{0, 1,\ldots, M\}$, and let $\mathbf{U}_q$ be the set of corresponding $q$-expansions. Recently, Komornik et al.~(Adv. Math., 2017) showed that the topological entropy function $H: q \mapsto h_{top}(\mathbf{U}_q)$ is a Devil's staircase in $(1, M+1]$. Let $\mathcal{B}$ be the bifurcation set of $H$ defined by \[ \mathcal{B}=\{q\in(1, M+1]: H(p)\ne H(q)\quad\textrm{for any}\quad p\ne q\}. \] In this paper we analyze the fractal properties of $\mathcal{B}$, and show that for any $q\in \mathcal{B}$, \[ \lim_{\delta\rightarrow 0} \dim_H(\mathcal{B}\cap(q-\delta, q+\delta))=\dim_H\mathcal{U}_q, \] where $\dim_H$ denotes the Hausdorff dimension. Moreover, when $q\in\mathcal{B}$ the univoque set $\mathcal{U}_q$ is dimensionally homogeneous, i.e., $ \dim_H(\mathcal{U}_q\cap V)=\dim_H\mathcal{U}_q $ for any open set $V$ that intersect $\mathcal{U}_q$. As an application we obtain a dimensional spectrum result for the set $\mathcal{U}$ containing all bases $q\in(1, M+1]$ such that $1$ admits a unique $q$-expansion. In particular, we prove that for any $t>1$ we have \[ \dim_H(\mathcal{U}\cap(1, t])=\max_{ q\le t}\dim_H\mathcal{U}_q. \] We also consider the variations of the sets $\mathcal{U}=\mathcal{U}(M)$ when $M$ changes., Comment: 36 pages and 1 figure. To appear in Acta Arithmetica

Let $M$ be a positive integer and $q \in(1,M+1].$ We consider expansions of real numbers in base $q$ over the alphabet $\{0,\ldots, M\}$. In particular, we study the set $\mathcal{U}_{q}$ of real numbers with a unique $q$-expansion, and the set $\mathbf{U}_q$ of corresponding sequences. It was shown in (Komornik et al, 2017 Adv. Math.) that the function $H$, which associates to each $q\in(1, M+1]$ the topological entropy of $\mathcal{U}_q$, is a Devil's staircase. In this paper we explicitly determine the plateaus of $H$, and characterize the bifurcation set $\mathcal E$ of $q$'s where the function $H$ is not locally constant. Moreover, we show that $\mathcal E$ is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift $(\mathbf{V}_q, \sigma),$ which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension of $\mathcal{U}_q$ coincide for all $q\in(1,M+1]$., Comment: 56 pages, 7 figures. To appear in Trans. Amer. Math. Soc

Given a positive integer $M$ and a real number $q >1$, a \emph{$q$-expansion} of a real number $x$ is a sequence $(c_i)=c_1c_2\cdots$ with $(c_i) \in \{0,\ldots,M\}^\infty$ such that \[x=\sum_{i=1}^{\infty} c_iq^{-i}.\] It is well known that if $q \in (1,M+1]$, then each $x \in I_q:=\left[0,M/(q-1)\right]$ has a $q$-expansion. Let $\mathcal{U}=\mathcal{U}(M)$ be the set of \emph{univoque bases} $q>1$ for which $1$ has a unique $q$-expansion. The main object of this paper is to provide new characterizations of $\mathcal{U}$ and to show that the Hausdorff dimension of the set of numbers $x \in I_q$ with a unique $q$-expansion changes the most if $q$ "crosses" a univoque base. Denote by $\mathcal{B}_2=\mathcal{B}_2(M)$ the set of $q \in (1,M+1]$ such that there exist numbers having precisely two distinct $q$-expansions. As a by-product of our results, we obtain an answer to a question of Sidorov (2009) and prove that \[\dim_H(\mathcal{B}_2\cap(q',q'+\delta))>0\quad\textrm{for any}\quad \delta>0,\] where $q'=q'(M)$ is the Komornik-Loreti constant., Comment: 16 pages. To appear in Monatshefte fur Mathematik (2017)

To each $\alpha\in(1/3,1/2)$ we associate the Cantor set $$\Gamma_{\alpha}:=\Big\{\sum_{i=1}^{\infty}\epsilon_{i}\alpha^i: \epsilon_i\in\{0,1\},\,i\geq 1\Big\}.$$ In this paper we consider the intersection $\Gamma_\alpha \cap (\Gamma_\alpha + t)$ for any translation $t\in\mathbb{R}$. We pay special attention to those $t$ with a unique $\{-1,0,1\}$ $\alpha$-expansion, and study the set $$D_\alpha:=\{\dim_H(\Gamma_\alpha \cap (\Gamma_\alpha + t)):t \textrm{ has a unique }\{-1,0,1\}\,\alpha\textrm{-expansion}\}.$$ We prove that there exists a transcendental number $\alpha_{KL}\approx 0.39433\ldots$ such that: $D_\alpha$ is finite for $\alpha\in(\alpha_{KL},1/2),$ $D_{\alpha_{KL}}$ is infinitely countable, and $D_{\alpha}$ contains an interval for $\alpha\in(1/3,\alpha_{KL}).$ We also prove that $D_\alpha$ equals $[0,\frac{\log 2}{-\log \alpha}]$ if and only if $\alpha\in (1/3,\frac {3-\sqrt{5}}{2}].$ As a consequence of our investigation we prove some results on the possible values of $\dim_{H}(\Gamma_\alpha \cap (\Gamma_\alpha + t))$ when $\Gamma_\alpha \cap (\Gamma_\alpha + t)$ is a self-similar set. We also give examples of $t$ with a continuum of $\{-1,0,1\}$ $\alpha$-expansions for which we can explicitly calculate $\dim_{H}(\Gamma_\alpha\cap(\Gamma_\alpha+t)),$ and for which $\Gamma_\alpha\cap (\Gamma_\alpha+t)$ is a self-similar set. We also construct $\alpha$ and $t$ for which $\Gamma_\alpha \cap (\Gamma_\alpha + t)$ contains only transcendental numbers. Our approach makes use of digit frequency arguments and a lexicographic characterisation of those $t$ with a unique $\{-1,0,1\}$ $\alpha$-expansion., Comment: 19 pages, two figures

In this paper we consider a general class $\mathcal E$ of self-similar sets with complete overlaps. Given a self-similar iterated function system $\Phi=(E, \{f_i\}_{i=1}^m)\in\mathcal E$ on the real line, for each point $x\in E$ we can find a sequence $(i_k)=i_1i_2\ldots\in\{1,\ldots,m\}^\mathbb N$, called a coding of $x$, such that $$ x=\lim_{n\to\infty}f_{i_1}\circ f_{i_{2}}\circ\cdots\circ f_{i_n}(0). $$ For $k=1,2,\ldots, \aleph_0$ or $2^{\aleph_0}$ we investigate the subset $\mathcal U_k(\Phi)$ which consists of all $x\in E$ having precisely $k$ different codings. Among several equivalent characterizations we show that $\mathcal U_1(\Phi)$ is closed if and only if $\mathcal U_{\aleph_0}(\Phi)$ is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of $\mathcal U_k(\Phi)$, and show that the corresponding Hausdorff measure of $\mathcal U_k(\Phi)$ is always infinite for any $k\ge 2$. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in $\mathcal U_k(\Phi)$ and ${U_{\aleph_0}(\Phi)}$., Comment: We add some new results in this version

Given a positive real number $x$, we consider the smallest base $q_s(x)\in(1,2)$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that \[ x=\sum_{i=1}^\infty\frac{d_i}{(q_s(x))^i}. \] In this paper we give complete characterizations of those $x$'s for which $q_s(x)\le q_{KL}$, where $q_{KL}$ is the Komornik-Loreti constant. Furthermore, we show that $q_s(x)=q_{KL}$ if and only if \[ x\in\left\{1, ~\frac{q_{KL}}{q_{KL}^2-1},~ \frac{1}{q_{KL}^2-1}, ~\frac{1}{q_{KL}(q_{KL}^2-1)}\right\}. \] Finally, we determine the explicit value of $q_s(x)$ if $q_s(x)

For $q>1$ we consider expansions in base $q$ over the alphabet $\{0,1,q\}$. Let $\mathcal{U}_q$ be the set of $x$ which have a unique $q$-expansions. For $k=2, 3,\cdots,\aleph_0$ let $\mathcal{B}_k$ be the set of bases $q$ for which there exists $x$ having $k$ different $q$-expansions, and for $q\in \mathcal{B}_k$ let $\mathcal{U}_q^{(k)}$ be the set of all such $x$'s which have $k$ different $q$-expansions. In this paper we show that \[ \mathcal{B}_{\aleph_0}=[2,\infty),\quad \mathcal{B}_k=(q_c,\infty)\quad \textrm{for any}\quad k\ge 2, \] where $q_c\approx 2.32472$ is the appropriate root of $x^3-3x^2+2x-1=0$. Moreover, we show that for any positive integer $k\ge 2$ and any $q\in\mathcal{B}_{k}$ the Hausdorff dimensions of $\mathcal{U}_q^{(k)}$ and $\mathcal{U}_q$ are the same, i.e., \[ \dim_H\mathcal{U}_q^{(k)}=\dim_H\mathcal{U}_q\quad\textrm{for any}\quad k\ge 2. \] Finally, we conclude that the set of $x$ having a continuum of $q$-expansions has full Hausdorff dimension., Comment: 15 page, to appear in Mathematische Zeitschrift

Given two positive integers $M$ and $k$, let $\B_k$ be the set of bases $q>1$ such that there exists a real number $x$ having exactly $k$ different $q$-expansions over the alphabet $\{0,1,\cdots,M\}$. In this paper we investigate the smallest base $q_2$ of $\B_2$, and show that if $M=2m$ the smallest base $$q_2 =\frac{m+1+\sqrt{m^2+2m+5}}{2},$$ and if $M=2m-1$ the smallest base $q_2$ is the appropriate root of $$ x^4=(m-1)\,x^3+2 m\, x^2+m \,x+1. $$ Moreover, for $M=2$ we show that $q_2$ is also the smallest base of $\B_k$ for all $k\ge 3$. This turns out to be different from that for $M=1$., Comment: 27 pages

Given a positive integer $M$ and a real number $q>1$, we consider the univoque set $\mathcal{U}_q$ of reals which have a unique $q$-expansion over the alphabet $\set{0,1,\cdots,M}$. In this paper we show that for any $x\in\mathcal{U}_q$ and all sufficiently small $\varepsilon>0$ the Hausdorff dimension $\dim_H\mathcal{U}_q\cap(x-\varepsilon, x+\varepsilon)$ equals either $\dim_H\mathcal{U}_q$ {or} zero. Moreover, we give a complete description of the typical points $x\in\mathcal{U}_q$ which satisfy \[ \dim_H\mathcal{U}_q\cap(x-\varepsilon, x+\varepsilon)=\dim_H\mathcal{U}_q\quad\textrm{for any}\quad \varepsilon>0, \] and prove that the set of typical points of $\mathcal{U}_q$ has full Hausdorff dimension. In particular, we show that if $\mathcal{U}_q$ is a Cantor set, then all points of $\mathcal{U}_q$ are typical points. This strengthen a result of de Vries and Komornik (Adv. Math., 2009)., Comment: This paper has been withdrawn by the author due to a crucial error in Theorem 3.5