Related papers: Coincidence of invariant measure for the alternate…

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 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…

We consider an independent and identically distributed (i.i.d.) random dynamical system of simple linear transformations on the unit interval $T_{\beta}(x)=\beta x$ (mod $1$), $x\in[0,1]$, $\beta>0$, which are the so-called…

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…

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 study invariant measures of a piecewise expanding map in $\mathbb{R}^m$ defined by an expanding similitude modulo lattice. Using the result of Bang on a problem of Tarski, we show that when the similarity ratio is not less than $m+1$, it…

The random beta-transformation K is isomorphic to a full shift. This relation gives an invariant measure for K that yields the Bernoulli convolution by projection. We study the local dimension of the invariant measure for K for special…

We consider a family S=S(a) of 2-valued transformations of special form on the segment [0,1] with measure $\mu=\int p(x) d\lambda$, which is absolutely continuous with respect to the Lebesgue measure $\lambda$. We endow S with a set of…

Let $\beta >1$ be a non-integer. We consider expansions of the form $\sum_{i=1}^{\infty} d_i \beta^{-i}$, where the digits $(d_i)_{i \geq 1}$ are generated by means of a Borel map $K_{\beta}$ defined on $\{0,1\}^{\N}\times [ 0, \lfloor…

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 $\beta$-transformation…

Classical result states that for two continuous and strict means $M,\,N \colon I^2 \to I$ ($I$ is an interval) there exists a unique $(M,N)$-invariant mean $K \colon I^2 \to I$, i.e. such a mean that $K \circ (M,N)=K$ and, moreover, the…

We study arithmetical and combinatorial properties of $\beta$-integers for $\beta$ being the root of the equation $x^2=mx-n, m,n \in \mathbb N, m \geq n+2\geq 3$. We determine with the accuracy of $\pm 1$ the maximal number of…

We study the problem of when, given a countable homogeneous structure $M$ and a space $S$ of expansions of $M$, every $\mathrm{Aut}(M)$-invariant probability measure on $S$ is exchangeable (i.e. invariant under all permutations of the…

Let $P$ be the set of all prime numbers, ${q_1},{q_2}, \cdots ,{q_m} \in P$, $P_k$ be the k-th $(k = 1,2, \cdots m)$ element of $P$ in ascending order of size, ${\alpha _1},{\alpha _2}, \cdots ,{\alpha _m}$ be positive integers, and ${\beta…

We introduce generalized $(\alpha,\beta)$-transformations, which include all $(\alpha,\beta)$ and generalized $\beta$-transformations, and prove that all transitive generalized $(\alpha,\beta)$-transformations satisfy the level-2 large…

Let $k$ be a nonnegative integer, and let $\alpha$ and $\beta$ be two permutations of $n$ symbols. We say that $\alpha$ and $\beta$ $k$-commute if $H(\alpha\beta, \beta\alpha)=k$, where $H$ denotes the Hamming metric between permutations.…

We construct a Lebesgue measure preserving natural extension of the random beta-transformation. This allows us to give a formula for the density of the absolutely continuous invariant probability measure, answering a question of Dajani and…

We construct Gaussian invariant measures for the two-dimensional Euler equation on the plane. We show the existence of solution with initial conditions in the support of the measures, namely $H^\beta_{loc}(\R^2)$ with $\beta<-1$. Uniqueness…

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 prove that whenever the selfmapping $(M_1,\dots,M_p)\colon I^p \to I^p$, ($p \in \mathbb{N}$ and $M_i$-s are $p$-variable means on the interval $I$) is invariant with respect to some continuous and strictly monotone mean $K \colon I^p…