Related papers: On the Beta Transformation
This article proposes a framework for the study of periodic maps $T$ from a (typically finite) set $X$ to itself when the set $X$ is equipped with one or more real- or complex-valued functions. The main idea, inspired by the time-evolution…
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:…
Consider the minimal $\beta$-shift containing the shift space generated by given Sturmian word. In this paper we characterize such $\beta$ and investigate its combinatorial, dynamical and topological properties and prove that such $\beta$…
This paper has two themes that are intertwined: The first is the dynamics of certain piecewise affine maps on the Euclidean space that arise from a class of analog-to-digital conversion methods called Sigma-Delta quantization. The second is…
There is a unique unitarily-invariant ensemble of $N\times N$ Hermitian matrices with a fixed set of real eigenvalues $a_1 > \dots > a_N$. The joint eigenvalue distribution of the $(N - 1)$ top-left principal submatrices of a random matrix…
Let $\Theta$ be a finite alphabet. We consider a bundle of measure preserving transformations $(T_{\theta})_{\theta \in \Theta}$ acting on a probability space $(X,\mu)$, which are chosen randomly according to an ergodic stochastic process…
For any transitive piecewise monotonic map for which the set of periodic measures is dense in the set of ergodic invariant measures (such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces), we show…
The beta-conjugates of a base of numeration $\beta > 1$, $\beta$ being a Parry number, were introduced by Boyd, in the context of the R\'enyi-Parry dynamics of numeration system and the beta-transformation. These beta-conjugates are…
Ergodic Optimization is the process of finding invariant probability measures that maximize the integral of a given function. It has been conjectured that "most" functions are optimized by measures supported on a periodic orbit, and it has…
We generalize the Umbral Calculus of G-C. Rota by studying not only sequences of polynomials and inverse power series, or even the logarithms studied in, but instead we study sequences of formal expressions involving the iterated logarithms…
First conformal transformations of the $S$-matrix are derived in massless $\phi^4$-theory. Then it is shown that the anomalous transformations can be rewritten as a symmetry once one has introduced a local coupling and interprets the charge…
We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the…
We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their $\beta$-expansions with the exponential order $\beta^{-n}$. Moreover, the Hausdorff dimensions of sets of the real numbers…
We completely solve ergodic optimization of a full shift with an uncountable alphabet $[0,1]$, which is one of the most well-known examples of infinite dimensional dynamical systems with positive mean dimension (and thus with infinite…
The beta-encoder was recently proposed as a quantization scheme for analog-to-digital conversion; in contrast to classical binary quantization, in which each analog sample x in [-1,1] is mapped to the first N bits of its base-2 expansion,…
In this paper, we build some ergodic theorems involving function $\Omega$, where $\Omega(n)$ denotes the number of prime factors of a natural number $n$ counted with multiplicities. As a combinatorial application, it is shown that for any…
A $\beta$-skeleton, $\beta \geq 1$, is a planar proximity undirected graph of an Euclidean points set, where nodes are connected by an edge if their lune-based neighbourhood contains no other points of the given set. Parameter $\beta$…
Let $\beta>1$. For $x \in [0,\infty)$, we have so-called the $\beta$-expansion of $x$ in base $\beta$ as follows: $$x= \sum_{j \leq k} x_{j}\beta^{j} = x_{k}\beta^{k}+ \cdots + x_{1}\beta+x_{0}+x_{-1}\beta^{-1} + x_{-2}\beta^{-2} + \cdots$$…
In this article we study the first return map defined on the switch region induced by the greedy and lazy maps. In particular we study the allowable sequences of return times, and when the first return map is a generalised L\"uroth series…
We consider a specific random graph which serves as a disordered medium for a particle performing biased random walk. Take a two-sided infinite horizontal ladder and pick a random spanning tree with a certain edge weight $c$ for the…