Related papers: Induced Random $\beta$-transformation
An aim of this article is to highlight dynamical differences between the greedy, and hence the lazy, $\beta$-shift (transformation) and an intermediate $\beta$-shift (transformation), for a fixed $\beta \in (1, 2)$. Specifically, a…
We show that every intermediate $\beta$-transformation is topologically conjugate to a greedy $\beta$-transformation with a hole at zero, and provide a counterexample illustrating that the correspondence is not one-to-one. This…
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…
The beta transformation is the iterated map $\beta x\,\mod1$; it generates the base-$\beta$ expansion of a real number x. Every iterated piece-wise monotonic map is topologically conjugate to the beta transformation. For all but a countable…
We prove that return time statistics of a dynamical system do not change if one passes to an induced (i.e. first return) map. We apply this to show exponential return time statistics in i) smooth interval maps with nowhere-dense critical…
For irreducible interval exchange transformations, we study the relation between the powers of induced map and the induced maps of powers and raise a condition of equivalence between them. And skew production of Rauzy induction map is set…
For ergodic measures we consider the return and entry times for a measure preserving transformation and its induced map on a positive measure subset. We then show that the limiting entry and return times distributions are the same for the…
We numerically study two conservative two-dimensional maps, namely the baker map (whose Lyapunov exponent is known to be positive), and a typical one (exhibiting a vanishing Lyapunov exponent) chosen from the generalized shift family of…
Assume that the interval $I=[0,1)$ is partitioned into finitely many intervals $I_1,\dots,I_r$ and consider a map $T\colon I\to I$ so that $T_{\vert I_s}$ is a translation for each $1 \le s \le r$. We do not assume that the images of these…
Generalized beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformation $x \mapsto \beta x ~\pmod 1$, where $\beta>1$, and replacing some of the branches with branches of constant…
In this paper, we investigate a class of non-invertible piecewise isometries on the upper half-plane known as Translated Cone Exchanges. These maps include a simple interval exchange on a boundary we call the baseline. We provide a…
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}…
We study discrete-time random walks on arbitrary networks with first-passage resetting processes. To the end, a set of nodes are chosen as observable nodes, and the walker is reset instantaneously to a given resetting node whenever it hits…
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…
A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences $x_{n+1}=F(x_n)$ generated by such maps display…
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…
A standard interval exchange map is a one-to-one map of the interval which is locally a translation except at finitely many singularities. We define for such maps, in terms of the Rauzy-Veech continuous fraction algorithm, a diophantine…
We study returns in dynamical systems: when a set of points, initially populating a prescribed region, swarms around phase space according to a deterministic rule of motion, we say that the return of the set occurs at the earliest moment…
For $i = 0, 1, 2, \dots, k$, let $\mu_i$ be a Borel probability measure on $[0,1]$ which is equivalent to Lebesgue measure $\lambda$ and let $T_i:[0,1] \rightarrow [0,1]$ be $\mu_i$-preserving ergodic transformations. We say that…
A typical interval exchange transformation has an infinite sequence of matrices associated to it by successive iterations of Rauzy induction. In 2010, W. A. Veech answered a question of A. Bufetov by showing that the interval exchange…