相关论文: Chaotic sets and Hausdorff dimension for L\"uroth …
L\"uroth series, like regular continued fractions, provide an interesting identification of real numbers with infinite sequences of integers. These sequences give deep arithmetic and measure-theoretic properties of subsets of numbers…
In this paper we consider Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including $A$-coupled expanding systems. We prove that Li-Yorke pairs of $A$- coupled-expanding system under some conditions have…
We show that for some simple classical chaotic dynamical systems the set of Li-Yorke pairs has full Hausdorff dimension on invariant sets.
We address the problem of determining the Hausdorff dimension of sets consisting of complex irrationals whose complex continued fraction digits satisfy prescribed restrictions and growth conditions. For the Hurwitz continued fraction, we…
In 1928, Jarn\'{\i}k \cite{Jar} obtained that the set of continued fractions with bounded coefficients has Hausdorff dimension one. Good \cite{Goo} observed a dimension drop phenomenon by proving that the Hausdorff dimension of the set of…
We prove that there exists a scrambled set for the Gauss map with full Hausdorff dimension. Meanwhile, we also investigate the topological properties of the sets of points with dense or non-dense orbits.
We construct a multiply Xiong chaotic set with full Hausdorff dimension everywhere that is contained in some multiply proximal cell for the full shift over finite symbols and the Gauss system respectively.
Following in the footsteps of P. Erd\H{o}s and A. R\'enyi we compute the Hausdorff dimension of sets of numbers whose digits with respect to their $Q$-Cantor series expansions satisfy various statistical properties. In particular, we…
In a recent paper [Adv. Math. 305:165--196, 2017], Komornik et al.~proved a long-conjectured formula for the Hausdorff dimension of the set $\mathcal{U}_q$ of numbers having a unique expansion in the (non-integer) base $q$, and showed that…
We investigate the Hausdorff dimension of level sets defined by digit growth rates in $\theta$-expansions, a generalization of regular continued fractions. For any $\alpha \geq 0$, we prove that the set \[ E_\theta(\alpha) = \left\{ x \in…
In this paper, we study the metrical theory of the growth rate of digits in L\"{u}roth expansions. More precisely, for $ x\in \left( 0,1 \right] $, let $ \left[ d_1\left( x \right) ,d_2\left( x \right) ,\cdots \right] $ denote the…
We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the…
We extend results of parametric geometry of numbers to a general diagonal flow on the space of lattices. Moreover, we compute the Hausdorff dimension of the set of trajectories with every given behavior, with respect to a nonstandard metric…
The Hausdorff dimensions of certain sets of real numbers described in terms of the \alpha-L\"uroth expansion are given.
We develop the geometry of Hurwitz continued fractions, a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. Based on a thorough study of the geometric properties of Hurwitz continued…
In the present paper the classical ideas of Hausdorff and Lebesgue are combined and the Hausdorff--Lebesgue measure is introduced. This makes it possible to obtain new results in chaotic dynamics.
We prove a quantitative distortion theorem for iterated function systems that generate sets of continued fractions. As a consequence, we obtain upper and lower bounds on the Hausdorff dimension of any set of real or complex continued…
To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number theoretical free energy…
We study the decomposition of real numbers into sums of L\"uroth sets, which are defined by numbers whose L\"uroth expansions have prescribed digit constraints. We establish several results on the congruence modulo 1 of sums of L\"uroth…
We investigate the dynamics of continued fractions and explore the ergodic behaviour of the products of mixed partial quotients in continued fractions of real numbers. For any function $\Phi:\mathbb N\to [2,+\infty)$ and any integer $d\geq…