相关论文: A natural extension for the greedy beta-transforma…

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…

We introduce a parametrised family of maps $\{S_{\eta}\}_{\eta \in [1,2]}$, called symmetric doubling maps, defined on $[-1,1]$ by $S_\eta (x)=2x-d\eta$, where $d\in \{-1,0,1 \}$. Each map $S_\eta$ generates binary expansions with digits…

A beta expansion is the analogue of the base 10 representation of a real number, where the base may be a non-integer. Although the greedy beta expansion of 1 using a non-integer base is in general infinitely long and non-repeating, it is…

We show that for any ergodic Lebesgue measure preserving transformation $f: [0,1) \rightarrow [0,1)$ and any decreasing sequence $\{b_i\}_{i=1}^{\infty}$ of positive real numbers with divergent sum, the set…

We consider positional numeration systems with negative real base $-\beta$, where $\beta>1$, and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of…

Given a pair $Q=(q_0,q_1)\in(1,\infty)^2$ with $q_0+q_1\ge q_0q_1$, a sequence $(c_i)\in\set{0,1}^\infty$ is called a $Q$-expansion of $x$ if<br/>\begin{equation*}<br/>x=\sum_{i=1}^{\infty}\frac{c_i}{q_{c_1}\cdots…

We study the typical growth rate of the number of words of length n which can be extended to beta-expansions of x. In the general case we give a lower bound for the growth rate, while in the case that the Bernoulli convolution associated to…

In this paper we consider continued fraction (CF) expansions on intervals different from $[0,1]$. For every $x$ in such interval we find a CF expansion with a finite number of possible digits. Using the natural extension, the density of the…

Let $ \beta $ be a real number less than -1. In this paper, we prove the uniqueness of the measure with maximal entropy of the negative $\beta$-shift. Endowed with the shift, this symbolic dynamical system is coded under certain conditions,…

We improve and subsume the conditions of Johansson and \"Oberg [18] and Berbee [2] for uniqueness of a g-measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique g-measures have…

We study the negative beta transformations $T_{-\beta}:=-\beta x +\lfloor\beta x\rfloor+1$ for $x\in(0,1]$ and $\beta>1$. We present a complete characterization of pairs of dstinct non-integers with the same $T_{-\beta}$-invariant measure:…

We define a random walk adic transformation associated to an aperiodic random walk on $G=\mathbb{Z}^{k}\times\mathbb{R}^{D-k}$ driven by a $\beta$-transformation and study its ergodic properties. In particular, this transformation is…

We compute explicitly the density of the invariant measure for the Reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira. We also apply the same method on the…

We construct a natural invariant measure concentrated on the set of square-free numbers, and invariant under the shift. We prove that the corresponding dynamical system is isomorphic to a translation on a compact, Abelian group. This…

Erd\H{o}s and Graham found it conceivable that the best $n$-term Egyptian underapproximation of almost every positive number for sufficiently large $n$ gets constructed in a greedy manner, i.e., from the best $(n-1)$-term Egyptian…

In this paper we study the set of digit frequencies that are realised by elements of the set of $\beta$-expansions. The main result of this paper demonstrates that as $\beta$ approaches $1,$ the set of digit frequencies that occur amongst…

We introduce infinitesimal weak containment for measure-preserving actions of a countable group $\Gamma$: an action $(X,\mu)$ is infinitesimally contained in $(Y,\nu)$ if the statistics of the action of $\Gamma$ on small measure subsets of…

We study the selfsimilarity and the Gibbs properties of several measures defined on the product space $\Omega\_r:=\{0,1,...,\break r-1\}^{\mathbb N}$. This space can be identified with the interval $[0,1]$ by means of the numeration in base…

We give a heuristic method to solve explicitly for an absolutely continuous invariant measure for a piecewise differentiable, expanding map of a compact subset $I$ of Euclidean space $R^d$. The method consists of constructing a skew product…

Ab\'ert-Weiss have shown that the Bernoulli shift s of a countably infinite group \Gamma is weakly contained in any free measure preserving action (mpa) b of \Gamma. We establish a strong version of this result, conjectured by Ioana, by…