arXiv subjects

Jun Jason Luo

Publications and source records attributed to Jun Jason Luo.

At least 19 recordsLinked to original sources

arXiv2026

Local Scaling and Dimension Distortion of Generalized Cantor Functions

Let $\mu$ be a self-similar Cantor measure on $\mathbb R$ associated with a probability weight vector $\mathbf p$, let $K=\operatorname{supp}\mu$, and let $F$ denote the distribution function of $\mu$. We characterize the points $x\in K$ at which the local scaling exponent \[ \lim_{\substack{y\to x, y\in K}} \frac{\log |F(y)-F(x)|}{\log |y-x|} \] exists and assumes a prescribed value. The characterization is formulated in terms of the convergence of the ratio between the accumulated logarithmic mass and geometric scales, together with the sublinear growth of the endpoint runs. Unlike the classical ternary case, our approach applies to arbitrary contraction ratios and probability weights. As an application, we construct a subset $M\subset K$ of full Hausdorff measure on which the local scaling exponent of $F$ is ${h(\mathbf q,\mathbf p)}/{\chi(\mathbf q)}$ and establish the exact dimension-distortion formula \[ \dim_{\mathrm H} F(A) = \frac{\chi(\mathbf q)}{h(\mathbf q,\mathbf p)} \dim_{\mathrm H} A \] for every $A\subset M$, where $\mathbf q$ is the natural probability vector, $\chi(\mathbf q)$ is the corresponding Lyapunov exponent, and $h(\mathbf q,\mathbf p)$ is the cross-entropy of $\mathbf q$ relative to $\mathbf p$. A three-branch example illustrates the results.

math.CA↗

arXiv2026

On the Hausdorff dimension of graph of random vector-valued Weierstrass function

Let $\Theta=\{\theta_n\}, \Lambda=\{\lambda_n\}$ be two sequences of independent and identically distributed uniform random variables on $[0,1]$. The random vector-valued Weierstrass function is given by $$ f_{\Theta,\Lambda}(x)= \left( \sum_{n=0}^{\infty} a^n\cos\bigl(2\pi (b^n x+\theta_n)\bigr),\ \sum_{n=0}^{\infty} a^n\sin\bigl(2\pi (b^n x+\lambda_n)\bigr) \right), \; x\in[0,1], $$ where $0 1$. The Hausdorff dimension of the graph of this function is proved to be $$\dim_H G(f_{\Theta,\Lambda}) = \min\left\{-\frac{\log b}{\log a}, \, 3 +2\frac{\log a}{\log b}\right\} \quad \text{a.s.}$$

math.CA↗

arXiv2025

Spectrality of a class of moran measures on $\mathbb{R}^2$

We investigate spectral properties of planar Moran measures $μ_{\{M_n\},\{D_n\}}$ generated by sequences of expanding matrices $\{M_n\}\subset GL(2,\mathbb{Z})$ and digit sets $\{D_n\}\subset\mathbb{Z}^2$, where each digit set has the form $$ D_n = \left\{ \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} α_{n_1} \\ α_{n_2} \end{pmatrix}, \begin{pmatrix} β_{n_1} \\ β_{n_2} \end{pmatrix}, \begin{pmatrix} -α_{n_1}-β_{n_1} \\ -α_{n_2}-β_{n_2} \end{pmatrix} \right\} $$ satisfying $α_{n_1}β_{n_2}-α_{n_2}β_{n_1} \ne 0 \pmod{2}$. Under the hypotheses $|\det(M_n)| > 4$ for all $n\geq 1$, $\sup_{n\geq 1}\|M_n^{-1}\| < 1$, and $\{D_n\}$ is finite, we establish the following characterization: $$ μ_{\{M_n\},\{D_n\}} \text{ is a spectral measure} \Longleftrightarrow M_n \in GL(2,2\mathbb{Z}) \text{ for all } n\geq 2. $$ Furthermore, for the critical case $|\det(M_n)| = 4$, we derive a complete spectral criterion for a significant class of Moran measures through combinatorial analysis of digit sets. These results extend current understanding of spectral self-affine measures to Moran-type constructions.

math.FA↗

arXiv2024

Spectrality of alternating-sign Moran measures

For $m\geq 2$, let $D_m=\{0,1,\ldots,m-1\}$. For each $k\geq 1$, consider the family of contractions \[ \Phi_k=\{\phi_{k,d}:d\in D_{n_k}\}, \qquad \phi_{k,d}(x)=(-1)^d b_k^{-1}(x+d), \] where $b_k$ and $n_k$ are integers satisfying $b_k\geq n_k\geq 2$. We construct the canonical pullback attractor generated by $\{\Phi_k\}_{k\geq 1}$ and the associated equal-weight Moran measure $\mu$. Suppose that $\{b_k\}_{k\geq 1}$ is bounded and that $n_k$ is even for every $k$. We prove that $\mu$ is spectral if $2\mid b_2,\ n_2\mid 2b_2$, and \(n_k\mid b_k$ for all $k\geq 3$. If, in addition, $4\mid n_k$ for every $k$, then the converse also holds, yielding a complete characterization of spectrality. The proof reduces the associated matrix-valued Fourier recursion to the Fourier transform of an infinite convolution of discrete probability measures. The main new ingredient in the necessity argument is a decomposition method for rational-scale infinite convolutions, in which a spectrum is successively partitioned into congruence classes.

math.FA↗

arXiv2017

On the Lipschitz equivalence of self-affine sets

Let $A$ be an expanding $d\times d$ matrix with integer entries and ${\mathcal D}\subset {\mathbb Z}^d$ be a finite digit set. Then the pair $(A, {\mathcal D})$ defines a unique integral self-affine set $K=A^{-1}(K+{\mathcal D})$. In this paper, by replacing the Euclidean norm with a pseudo-norm $w$ in terms of $A$, we construct a hyperbolic graph on $(A, {\mathcal D})$ and show that $K$ can be identified with the hyperbolic boundary. Moreover, if $(A, {\mathcal D})$ safisfies the open set condition, we also prove that two totally disconnected integral self-affine sets are Lipschitz equivalent if an only if they have the same $w$-Hausdorff dimension, that is, their digit sets have equal cardinality. We extends some well-known results in the self-similar sets to the self-affine sets.

math.GT↗

arXiv2017

Lipschitz equivalence of Cantor sets and irreducibility of polynomials

In the paper, we provide an effective method for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the contraction vectors is homogeneous.

math.GT↗

arXiv2017

Topological properties of self-similar fractals with one parameter

In this paper, we study two classes of planar self-similar fractals $T_\varepsilon$ with a shifting parameter $\varepsilon$. The first one is a class of self-similar tiles by shifting $x$-coordinates of some digits. We give a detailed discussion on the disk-likeness ({\it i.e., the property of being a topological disk}) in terms of $\varepsilon$. We also prove that $T_\varepsilon$ determines a quasi-periodic tiling if and only if $\varepsilon$ is rational. The second one is a class of self-similar sets by shifting diagonal digits. We give a necessary and sufficient condition for $T_\varepsilon$ to be connected.

math.GN↗

arXiv2016

Self-similar sets, simple augmented trees, and their Lipschitz equivalence

Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graph on the IFS has been established recently. In the paper, we introduce a notion of simple augmented tree which is a Gromov hyperbolic graph. By generalizing a combinatorial device of rearrangeable matrix, we show that there exists a near-isometry between the simple augmented tree and the symbolic space of the IFS, so that their hyperbolic boundaries are Lipschitz equivalent. We then apply this to consider the Lipschitz equivalence of self-similar sets with or without assuming the open set condition. Moreover, we also provide a criterion for a self-similar set to be a Cantor-type set which completely answers an open question raised in \cite{LaLu13}. Our study extends the previous works.

math.GT↗

arXiv2016

A characterization of connected self-affine fractals arising from collinear digits

Let $A$ be an expanding integer matrix with characteristic polynomial $f(x)=x^{2}+px+q$, and let $\mathcal{D}=\{0,1,\dots,|q|-2,|q|+m\}\mathbf{v}$ be a collinear digit set where $m\geqslant 0, {\mathbf v}\in {\mathbb Z}^2$. It is well known that there exists a unique self-affine fractal $T$ satisfying $AT=T+\mathcal{D}$. In this paper, we give a complete characterization on the connected $T$. That generalizes the previous result of $|q|=3$.

math.GN↗

arXiv2016

Spectral property of self-affine measures on ${\mathbb R}^n$

We study spectral properties of the self-affine measure $μ_{M,\mathcal {D}}$ generated by an expanding integer matrix $M\in M_n(\mathbb{Z})$ and a consecutive collinear digit set $\mathcal {D}=\{0,1,\dots,q-1\}v$ where $v\in \mathbb{Z}^n\setminus\{0\}$ and $q\ge 2$ is an integer. Some sufficient conditions for $μ_{M,\mathcal {D}}$ to be a spectral measure or to have infinitely many orthogonal exponentials are given. Moreover, for some special cases, we can obtain a necessary and sufficient condition on the spectrality of $μ_{M,\mathcal {D}}$. Our study generalizes the one dimensional results proved by Dai, {\it et al.} (\cite{Dai-He-Lai_2013, Dai-He-Lau_2014}).

math.CA↗

arXiv2016

Connectedness of self-affine sets with product digit sets

Let $T(A,\mathcal{D})$ be a self-affine set generated by an expanding matrix $A=\left[\begin{array}{rr} p & 0\cr -a & q \end{array}\right]$ and a product digit set $\mathcal{D}=\{0,1,\dots,m-1\}\times \{0,1,\dots,n-1\}$. We provide a necessary and sufficient condition for the $T(A,\mathcal{D})$ to be connected, which generalizes the known results.

math.GN↗

arXiv2015

On the classification of fractal squares

In \cite{LaLuRa13}, the authors completely classified the topological structure of so called {\it fractal square} $F$ defined by $F=(F+{\mathcal D})/n$, where ${\mathcal{D}}\subsetneq\{0,1,\dots,n-1\}^2, n\ge 2$. In this paper, we further provide simple criteria for the $F$ to be totally disconnected, then we discuss the Lipschitz classification of $F$ in the case $n=3$, which is an attempt to consider non-totally disconnected sets.

math.GN↗

arXiv2014

Lipschitz equivalence of self-similar sets and hyperbolic boundaries II

In \cite{LuLa13}, two of the authors initiated a study of Lipschitz equivalence of self-similar sets through the augmented trees, a class of hyperbolic graphs introduced by Kaimanovich \cite{Ka03} and developed by Lau and Wang \cite{LaWa09}. In this paper, we continue such investigation. We remove a major assumption in the main theorem in \cite{LuLa13} by using a new notion of quasi-rearrangeable matrix, and show that the hyperbolic boundary of any simple augmented tree is Lipschitz equivalent to a Cantor-type set. We then apply this result to consider the Lipschitz equivalence of certain totally disconnected self-similar sets as well as their unions.

math.CO↗

arXiv2014

On the connectedness of planar self-affine sets

In this paper, we consider the connectedness of planar self-affine set $T(A,\mathcal{D})$ arising from an integral expanding matrix $A$ with characteristic polynomial $f(x)=x^2+bx+c$ and a digit set $\mathcal{D}=\{0,1,\dots, m\}v$. The necessary and sufficient conditions only depending on $b,c,m$ are given for the $T(A,\mathcal{D})$ to be connected. Moreover, we also consider the case that ${\mathcal D}$ is non-consecutively collinear.

math.DS↗

arXiv2013

Boundaries of Disk-like Self-affine Tiles

Let $T:= T(A, {\mathcal D})$ be a disk-like self-affine tile generated by an integral expanding matrix $A$ and a consecutive collinear digit set ${\mathcal D}$, and let $f(x)=x^{2}+px+q$ be the characteristic polynomial of $A$. In the paper, we identify the boundary $\partial T$ with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair $(A,{\mathcal D})$ to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm $ω$, we find the generalized Hausdorff dimension $\dim_H^ω (\partial T)=2\log ρ(M)/\log |q|$ where $ρ(M)$ is the spectral radius of certain contact matrix $M$. Especially, when $A$ is a similarity, we obtain the standard Hausdorff dimension $\dim_H (\partial T)=2\log ρ/\log |q|$ where $ρ$ is the largest positive zero of the cubic polynomial $x^{3}-(|p|-1)x^{2}-(|q|-|p|)x-|q|$, which is simpler than the known result.

math.MG↗

arXiv2012

Lipschitz equivalence of self-similar sets and hyperbolic boundaries

In [9] Kaimanovich introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown in [13] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently.

math.MG↗

arXiv2012

Topological Structure of Fractal Squares

Given an integer $n\geq 2$ and a digit set ${\mathcal D}\subsetneq {0,1,...,n-1}^2$, there is a self-similar set $F \subset {\Bbb R}^2$ satisfying the set equation: $F=(F+{\mathcal D})/n$. We call such $F$ a fractal square. By studying a periodic extension $H= F+ {\mathbb Z}^2$, we classify $F$ into three types according to their topological properties. We also provide some simple criteria for such classification.

math.GN↗

arXiv2012

Moran Sets and Hyperbolic Boundaries

In the paper, we prove that a Moran set is homeomorphic to the hyperbolic boundary of the representing symbolic space in the sense of Gromov, which generalizes the results of Lau and Wang [Indiana U. Math. J. {\bf 58} (2009), 1777-1795]. Moreover, by making use of this, we establish the Lipschitz equivalence of a class of Moran sets.

math.MG↗