Related papers: Dynamical properties of the negative beta transfor…

For any $\beta > 1$, let $T_\beta: [0,1)\rightarrow [0,1)$ be the $\beta$-transformation defined by $T_\beta x=\beta x \mod 1$. We study the uniform recurrence properties of the orbit of a point under the $\beta$-transformation to the point…

We introduce a family of dynamical systems that generate negative $\beta$-expansions and study the support of the invariant measure which is absolutely continuous with respect to Lebesgue measure. We give a characterization of the set of…

We consider base-$\beta$ expansions of Parry's type, where $a_0 \geq a_1 \geq 1$ are integers and $a_0<\beta <a_0+1$ is the positive solution to $\beta^2 = a_0\beta + a_1$ (the golden ratio corresponds to $a_0=a_1=1$). The map $x\mapsto…

We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution \[ \mu_\omega = \mathop{\circledast}_{k=1}^{\infty} \left( \frac{\delta_0 + \delta_{\lambda_1 \lambda_2 \ldots \lambda_{k-1}…

We determine the structure of the set of intermediate $\beta$-shifts of finite type. Specifically, we show that this set is dense in the parameter space $\Delta = \{ (\beta, \alpha) \in \mathbb{R}^{2} \colon \beta \in (1, 2) \; \text{and}…

The (-\beta)-integers are natural generalisations of the \beta-integers, and thus of the integers, for negative real bases. They can be described by infinite words which are fixed points of anti-morphisms. We show that they are not…

We consider a class of mass transfer models on a one-dimensional lattice with nearest-neighbour interactions. The evolution is given by the discrete backward fast diffusion equation, with exponent $\beta$ in the regime $(-\infty,0) \cup…

We present a complete characterization of two different non-integers with the same R\'{e}nyi-Parry measure. We prove that for two non-integers $\beta_1 ,\beta_2 >1$, the R\'{e}nyi-Parry measures coincide if and only if $\beta_1$ is the root…

In this paper, we study representations of real numbers in the positional numeration system with negative basis, as introduced by Ito and Sadahiro. We focus on the set $\Z_{-\beta}$ of numbers whose representation uses only non-negative…

The Thue--Morse sequence $\{t(n)\}_{n\geqslant 1}$ is the indicator function of the parity of the number of ones in the binary expansion of positive integers $n$, where $t(n)=1$ (resp. $=0$) if the binary expansion of $n$ has an odd (resp.…

For a real number $0<\lambda<2$, we introduce a transformation $T_\lambda$ naturally associated to expansion in $\lambda$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of…

We consider the random $\beta$-transformation $K_{\beta}$, defined on $\{0,1\}^{\mathbb N}\times[0, \frac{\lfloor\beta\rfloor}{\beta-1}]$, that generates all possible expansions of the form $x=\sum_{i=0}^{\infty}\frac{a_i}{\beta^i}$, where…

In the freezing regime where the system size N is fixed and the inverse temperature beta tends to infinity, the eigenvalues of Gaussian beta ensembles converge to zeros of the Nth Hermite polynomial. That law of large numbers has been…

We compare a piecewise linear map with constant slope beta>1 and a piecewise linear map with constant slope -beta. These maps are called the positive and negative beta-transformations. We show that for a certain set of beta's, the…

We show that every subshift factor of a ($-\beta$)-shift is intrinsically ergodic, when $\beta\geq \frac{1+\sqrt{5}}{2}$ and the ($-\beta$)-expansion of $1$ is not periodic with odd period. Moreover, the unique measure of maximal entropy…

In this paper we study digit frequencies in the setting of expansions in non-integer bases, and self-affine sets with non-empty interior. Within expansions in non-integer bases we show that if $\beta\in(1,1.787\ldots)$ then every…

In a recent paper of Feng and Sidorov they show that for $\beta\in(1,\frac{1+\sqrt{5}}{2})$ the set of $\beta$-expansions grows exponentially for every $x\in(0,\frac{1}{\beta-1})$. In this paper we study this growth rate further. We also…

Let $\Gamma $ be an infinite discrete group and $\mathsf{A}\subset \Gamma $ a nonempty finite subset. The set of permutations $\sigma $ of $\Gamma $ such that $s^{-1}\sigma (s)\in \mathsf{A}$ for every $s\in \Gamma $ can be identified with…

From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different…

For any $n\geq 3$, let $1<\beta<2$ be the largest positive real number satisfying the equation $$\beta^n=\beta^{n-2}+\beta^{n-3}+\cdots+\beta+1.$$ In this paper we define the shrinking random $\beta$-transformation $K$ and investigate…