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Lingmin Liao

Publications and source records attributed to Lingmin Liao.

At least 37 records · Page 2Linked to original sources
arXiv2022

Uniform approximation problems of expanding Markov maps

Let $ T:[0,1]\to[0,1] $ be an expanding Markov map with a finite partition. Let $ μ_ϕ$ be the invariant Gibbs measure associated with a Hölder continuous potential $ ϕ$. In this paper, we investigate the size of the uniform approximation set \[\mathcal U^κ(x):=\{y\in[0,1]:\forall N\gg1,~\exists n\le N, \text{ such that }|T^nx-y| 0 $ and $ x\in[0,1] $. The critical value of $ κ$ such that $ \textrm{dim}_{\textrm H}\mathcal U^κ(x)=1 $ for $ μ_ϕ$-a.e.$ \, x $ is proven to be $ 1/α_{\max} $, where $ α_{\max}=-\int ϕ\,dμ_{\max}/\int\log|T'|\,dμ_{\max} $ and $ μ_{\max} $ is the Gibbs measure associated with the potential $ -\log|T'| $. Moreover, when $ κ>1/α_{\max} $, we show that for $ μ_ϕ$-a.e.$ \, x $, the Hausdorff dimension of $ \mathcal U^κ(x) $ agrees with the multifractal spectrum of $ μ_ϕ$.

math.DS↗
arXiv2021

Big Birkhoff sums in $d$-decaying Gauss like iterated function systems

The increasing rate of the Birkhoff sums in the infinite iterated function systems with polynomial decay of the derivative (for example the Gauss map) is studied. For different unbounded potential functions, the Hausdorff dimensions of the sets of points whose Birkhoff sums share the same increasing rate are obtained.

math.NT↗
arXiv2021

Uniform random covering problems

Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence $ω=(ω_n)_{n\geq 1}$ uniformly distributed on the unit circle $\mathbb{T}$ and a sequence $(r_n)_{n\geq 1}$ of positive real numbers with limit $0$. We investigate the size of the random set \[ \mathcal U (ω):=\{y\in \mathbb{T}: \ \forall N\gg 1, \ \exists n \leq N, \ \text{s.t.} \ \| ω_n -y \| < r_N \}. \] Some sufficient conditions for $\mathcal U(ω)$ to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that $\mathcal U(ω)$ is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.

math.PR↗
arXiv2020

Odd-odd continued fraction algorithm

By using a jump transformation associated to the Romik map, we define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. Among others, it is proved that all the best approximating rationals of odd denominators and odd numerators of an irrational number are given by the principal convergents of the odd-odd continued fraction algorithm and vice versa.

math.DS↗
arXiv2019

Stationary determinantal processes: $ψ$-mixing property and $L^q$-dimensions

The results of this paper are 3-folded. Firstly, for any stationary determinantal process on the integer lattice, induced by strictly positive and strictly contractive involution kernel, we obtain the necessary and sufficient condition for the $ψ$-mixing property. Secondly, we obtain the existence of the $L^q$-dimensions of the stationary determinantal measure on symbolic space $\{0, 1\}^\mathbb{N}$ under appropriate conditions. Thirdly, the previous two results together imply the precise increasing rate of the longest common substring of a typical pair of points in $\{0, 1\}^\mathbb{N}$.

math.PR↗
arXiv2018

On the shortest distance between orbits and the longest common substring problem

In this paper, we study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions, the decay of the shortest distance depends on the correlation dimension. For irrational rotations, we prove a different behaviour depending on the irrational exponent of the angle of the rotation. For random processes, this problem corresponds to the longest common substring problem. We extend the result of Arratia and Waterman on sequence matching to $α$-mixing processes with exponential decay.

math.DS↗
arXiv2018

Hausdorff dimension of weighted singular vectors

Let $w=(w_1, w_2)$ be a pair of positive real numbers with $w_1+w_2=1$ and $w_1\ge w_2$. We show that the set of $w$-weighted singular vectors in $\mathbb R^2$ has Hausdorff dimension $2- \frac{1}{1+w_1}$. This extends the previous work of Yitwah Cheung on the Hausdorff dimension of the usual (unweighted) singular vectors in $\mathbb R^2$.

math.DS↗
arXiv2018

Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential

Let $((0,1], T)$ be the doubling map in the unit interval and $φ$ be the Saint-Petersburg potential, defined by $φ(x)=2^n$ if $x\in (2^{-n-1}, 2^{-n}]$ for all $n\geq 0$. We consider the asymptotic properties of the Birkhoff sum $S\_n(x)=φ(x)+\cdots+φ(T^{n-1}(x))$. With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that $\frac{1}{n\log n}S\_n(x)$ converges to $\frac{1}{\log 2}$ in probability. We determine the Hausdorff dimension of the level set $\{x: \lim\_{n\to\infty}S\_n(x)/n=α\} \ (α>0)$, as well as that of the set $\{x: \lim\_{n\to\infty}S\_n(x)/Ψ(n)=α\} \ (α>0)$, when $Ψ(n)=n\log n, n^a $ or $2^{n^γ}$ for $a>1$, $γ>0$. The fast increasing Birkhoff sum of the potential function $x\mapsto 1/x$ is also studied.

math.DS↗
arXiv2017

Metrical results on the distribution of fractional parts of powers of real numbers

Denote by {$\times$} the fractional part. We establish several new metrical results on the distribution properties of the sequence ({x n }) n$\ge$1. Many of them are presented in a more general framework, in which the sequence of functions (x $\rightarrow$ x n) n$\ge$1 is replaced by a sequence (fn) n$\ge$1 , under some growth and regularity conditions on the functions fn.

math.NT↗
arXiv2017

Dirichlet uniformly well-approximated numbers

Fix an irrational number $θ$. For a real number $τ>0$, consider the numbers $y$ satisfying that for all large number $Q$, there exists an integer $1\leq n\leq Q$, such that $\|nθ-y\| 0$, the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine property of $θ$. It is also proved that with respect to $τ$, the only possible discontinuous point of the Hausdorff dimension is $τ=1$.

math.NT↗
arXiv2017

Periodic $p$-adic Gibbs measures of $q$-states Potts model on Cayley tree: The chaos implies the vastness of $p$-adic Gibbs measures

We study the set of $p$-adic Gibbs measures of the $q$-states Potts model on the Cayley tree of order three. We prove the vastness of the periodic $p$-adic Gibbs measures for such model by showing the chaotic behavior of the correspondence Potts--Bethe mapping over $\mathbb{Q}\_p$ for $p\equiv 1 \ (\rm{mod} \ 3)$. In fact, for $0 < |θ-1|\_p < |q|\_p^2 < 1$, there exists a subsystem that isometrically conjugate to the full shift on three symbols. Meanwhile, for $0 < |q|\_p^2 \leq |θ-1|\_p < |q|\_p < 1$, there exists a subsystem that isometrically conjugate to a subshift of finite type on $r$ symbols where $r \geq 4$. However, these subshifts on $r$ symbols are all topologically conjugate to the full shift on three symbols. The $p$-adic Gibbs measures of the same model for the cases $p=2,3$ and the corresponding Potts--Bethe mapping are also discussed.Furthermore, for $0 < |θ-1|\_p < |q|\_p < 1,$ we remark that the Potts--Bethe mapping is not chaotic when $p=2,\ p=3$ and $p\equiv 2 \ (\rm{mod} \ 3)$ and we could not conclude the vastness of the periodic $p$-adic Gibbs measures. In a forthcoming paper with the same title, we will treat the case $0 < |q|\_p \leq |θ-1|\_p < 1$ for all $p$.

math.DS↗
arXiv2016

Minimality of p-adic rational maps with good reduction

A rational map with good reduction in the field $\mathbb{Q}\_p$ of $p$-adic numbers defines a $1$-Lipschitz dynamical system on the projective line $\mathbb{P}^1(\mathbb{Q}\_p)$ over $\mathbb{Q}\_p$. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, $\mathbb{P}^1(\mathbb{Q}\_p)$ is decomposed into three parts: finitely many periodic orbits; finite or countably many minimal subsystems each consisting of a finite union of balls; and the attracting basins of periodic orbits and minimal subsystems. For any prime $p$, a criterion of minimality for rational maps with good reduction is obtained. When $p=2$, a condition in terms of the coefficients of the rational map is proved to be necessary for the map being minimal and having good reduction, and sufficient for the map being minimal and $1$-Lipschitz. It is also proved that a rational map having good reduction of degree $2$, $3$ and $4$ can never be minimal on the whole space $\mathbb{P}^1(\mathbb{Q}\_2)$.

math.DS↗
arXiv2015

Fuglede's conjecture holds in $\mathbb{Q}_p$

Fuglede's conjecture in $\mathbb{Q}_p$ is proved. That is to say, a Borel set of positive and finite Haar measure in $\mathbb{Q}_p$ is a spectral set if and only if it tiles $\mathbb{Q}_p$ by translation.

math.CA↗