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De-Jun Feng

Publications and source records attributed to De-Jun Feng.

At least 19 recordsLinked to original sources
arXiv2025

Distinct dimensions for attractors of bi-Lipschitz iterated function systems

In this paper, we construct an iterated function system on the line consisting of two bi-Lipschitz contractions whose attractor has distinct lower, Hausdorff, lower box, upper box, and Assouad dimensions, thereby providing negative answers to certain folklore questions. Furthermore, as a by-product of our study of bi-Lipschitz IFSs, we construct IFSs within this family that exhibit interesting fractal behaviour. In particular, we prove the following two statements: (i) There exists a bi-Lipschitz IFS for which the pushforward of any ergodic measure with positive entropy is not exact dimensional; (ii) There exists a bi-Lipschitz IFS whose attractor has empty interior yet positive Lebesgue measure.

math.DS↗
arXiv2025

Dimensions of orthogonal projections of typical self-affine sets and measures

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in {\Bbb R}^{md}$, let $\pi^{\bf a}\colon \Sigma=\{1,\ldots, m\}^{\Bbb N}\to {\Bbb R}^d$ denote the coding map associated with the affine IFS $\{T_ix+a_i\}_{i=1}^m$, and let $K^{\bf a}$ denote the attractor of this IFS. Let $W$ be a linear subspace of ${\Bbb R}^d$ and $P_W$ the orthogonal projection onto $W$. We show that for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$, the Hausdorff and box-counting dimensions of $P_W(K^{\bf a})$ coincide and are determined by the zero point of a certain pressure function associated with $T_1,\ldots, T_m$ and $W$. Moreover, for every ergodic $\sigma$-invariant measure $\mu$ on $\Sigma$ and for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$, the local dimensions of $(P_W\pi^{\bf a})_*\mu$ exist almost everywhere, here $(P_W\pi^{\bf a})_*\mu$ stands for the push-forward of $\mu$ by $P_W\pi^{\bf a}$. However, as illustrated by examples, $(P_W\pi^{\bf a})_*\mu$ may not be exact dimensional for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$. Nevertheless, when $\mu$ is a Bernoulli product measure, or more generally, a supermultiplicative ergodic $\sigma$-invariant measure, $(P_W\pi^{\bf a})_*\mu$ is exact dimensional for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$.

math.DS↗
arXiv2024

Dimension of homogeneous iterated function systems with algebraic translations

Let $ μ$ be the self-similar measure associated with a homogeneous iterated function system $ Φ= \{ λx + t_j \}_{j=1}^m $ on ${\Bbb R}$ and a probability vector $ (p_{j})_{j=1}^m$, where $0\neq λ\in (-1,1)$ and $t_j\in {\Bbb R}$. Recently by modifying the arguments of Varjú (2019), Rapaport and Varjú (2024) showed that if $t_1,\ldots, t_m$ are rational numbers and $0<λ<1$, then $$ \dim μ=\min\Big \{ 1, \; \frac{\sum_{j=1}^m p_{j}\log p_{j}}{ \log |λ| }\Big\}$$ unless $ Φ$ has exact overlaps. In this paper, we further show that the above equality holds in the case when $t_1,\ldots, t_m$ are algebraic numbers and $0<|λ|<1$. This is done by adapting and extending the ideas employed in the recent papers of Breuillard, Rapaport and Varjú.

math.DS↗
arXiv2023

Dimensions of projected sets and measures on typical self-affine sets

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in \Bbb R^{md}$, let $π^{\bf a}:\; Σ=\{1,\ldots, m\}^{\Bbb N}\to \Bbb R^d$ denote the coding map associated with the affine IFS $\{T_ix+a_i\}_{i=1}^m$. We show that for every Borel probability measure $μ$ on $Σ$, each of the following dimensions (lower and upper Hausdorff dimensions, lower and upper packing dimensions) of $π^{\bf a}_*μ$ is constant for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$, where $π^{\bf a}_*μ$ stands for the push-forward of $μ$ by $π^{\bf a}$. In particular, we give a necessary and sufficient condition on $μ$ so that $π^{\bf a}_*μ$ is exact dimensional for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$. Moreover, for every analytic set $E\subset Σ$, each of the Hausdorff, packing, lower and upper box-counting dimensions of $π^{\bf a}(E)$ is constant for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$. Formal dimension formulas of these projected measures and sets are given. The Hausdorff dimensions of exceptional sets are estimated.

math.DS↗
arXiv2022

Typical self-affine sets with non-empty interior

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\| <1/2$ for $1\leq i\leq m$. We provide some sufficient conditions on these matrices such that the self-affine set generated by the iterated function system $\{T_ix+a_i\}_{i=1}^m$ on $\Bbb R^d$ has non-empty interior for almost all $(a_1,\ldots, a_m)\in \Bbb R^{md}$.

math.CA↗
arXiv2022

A separation property for iterated function systems of similitudes

Let $E$ be the attractor of an iterated function system $\{ϕ_i(x)=ρR_ix+a_i\}_{i=1}^N$ on $\Bbb R^d$, where $0<ρ<1$, $a_i\in \Bbb R^d$ and $R_i$ are orthogonal transformations on $\Bbb R^d$. Suppose that $\{ϕ_i\}_{i=1}^N$ satisfies the open set condition, but not the strong separation condition. We show that $E$ can not be generated by any iterated function system of similitudes satisfying the strong separation condition. This gives a partial answer to a folklore question about the separation conditions on the generating iterated function systems of self-similar sets.

math.DS↗
arXiv2021

Estimates on the dimension of self-similar measures with overlaps

In this paper, we provide an algorithm to estimate from below the dimension of self-similar measures with overlaps. As an application, we show that for any $ β\in(1,2) $, the dimension of the Bernoulli convolution $ μ_β$ satisfies \[ \dim (μ_β) \geq 0.9804085,\] which improves a previous uniform lower bound $0.82$ obtained by Hare and Sidorov \cite{HareSidorov2018}. This new uniform lower bound is very close to the known numerical approximation $ 0.98040931953\pm 10^{-11}$ for $\dim μ_{β_3}$, where $ β_{3} \approx 1.839286755214161$ is the largest root of the polynomial $ x^{3}-x^{2}-x-1$. Moreover, the infimum $\inf_{β\in (1,2)}\dim (μ_β) $ is attained at a parameter $β_*$ in a small interval \[ (β_{3} -10^{-8}, β_{3} + 10^{-8}).\] When $β$ is a Pisot number, we express $\dim(μ_β)$ in terms of the measure-theoretic entropy of the equilibrium measure for certain matrix pressure function, and present an algorithm to estimate $\dim (μ_β)$ from above as well.

math.DS↗
arXiv2021

Dimension estimates for $C^1$ iterated function systems and repellers. Part II

This is the second part of our study of the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on ${\Bbb R}^d$. In the first part we proved that the upper box-counting dimension of the attractor of any $C^1$ IFS on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Here we introduce a generalized transversality condition (GTC) for parametrized families of $C^1$ IFSs, and show that these upper bounds give actually the dimensions for "typical" $C^1$ IFSs under this transversality condition. Moreover we verify the GTC for some parametrized families of $C^1$ IFSs on ${\Bbb R}^d$.

math.DS↗
arXiv2020

Dimensions of random statistically self-affine Sierpinski sponges in $\mathbb R^k$

We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge $K\subset \mathbb{R}^k$ ($k\ge 2$) obtained by using some percolation process in $[0,1]^k$. To do so, we first exhibit a Ledrappier-Young type formula for the Hausdorff dimensions of statistically self-affine measures supported on $K$. This formula presents a new feature compared to its deterministic or random dynamical version. Then, we establish a variational principle expressing $\dim_H K$ as the supremum of the Hausdorff dimensions of statistically self-affine measures supported on $K$, and show that the supremum is uniquely attained. The value of $\dim_H K$ is also expressed in terms of the weighted pressure function of some deterministic potential. As a by-product, when $k=2$, we give an alternative approach to the Hausdorff dimension of $K$, which was first obtained by Gatzouras and Lalley \cite{GL94}. The value of the box counting dimension of $K$ and its equality with $\dim_H K$ are also studied. We also obtain a variational formula for the Hausdorff dimensions of some orthogonal projections of $K$, and for statistically self-affine measures supported on~$K$, we establish a dimension conservation property through these projections.

math.DS↗
arXiv2020

Dimension estimates for $C^1$ iterated function systems and repellers. Part I

This is the first article in a two-part series containing some results on dimension estimates for $C^1$ iterated function systems and repellers. In this part, we prove that the upper box-counting dimension of the attractor of any $C^1$ iterated function system (IFS) on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Similar results are obtained for the repellers for $C^1$ expanding maps on Riemannian manifolds.

math.DS↗
arXiv2020

Dimension of invariant measures for affine iterated function systems

Let $\{S_i\}_{i\in Λ}$ be a finite contracting affine iterated function system (IFS) on ${\Bbb R}^d$. Let $(Σ,σ)$ denote the two-sided full shift over the alphabet $Λ$, and $π:Σ\to {\Bbb R}^d$ be the coding map associated with the IFS. We prove that the projection of an ergodic $σ$-invariant measure on $Σ$ under $π$ is always exact dimensional, and its Hausdorff dimension satisfies a Ledrappier-Young type formula. Furthermore, the result extends to average contracting affine IFSs. This completes several previous results and answers a folklore open question in the community of fractals. Some applications are given to the dimension of self-affine sets and measures.

math.DS↗
arXiv2020

On multifractal formalism for self-similar measures with overlaps

Let $μ$ be a self-similar measure generated by an IFS $Φ=\{ϕ_i\}_{i=1}^\ell$ of similarities on $\mathbb R^d$ ($d\ge 1$). When $Φ$ is dimensional regular (see Definition~1.1), we give an explicit formula for the $L^q$-spectrum $τ_μ(q)$ of $μ$ over $[0,1]$, and show that $τ_μ$ is differentiable over $(0,1]$ and the multifractal formalism holds for $μ$ at any $α\in [τ_μ'(1),τ_μ'(0+)]$. We also verify the validity of the multifractal formalism of $μ$ over $[τ_μ'(\infty),τ_μ'(0+)]$ for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds. For one of them, the proof appeals to a recent result of Shmerkin on the $L^q$-spectrum of self-similar measures.

math.DS↗
arXiv2018

On the Hausdorff Dimension of Bernoulli Convolutions

We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution $ν_β$ to arbitrary given accuracy whenever $β$ is algebraic. In particular, if the Garsia entropy $H(β)$ is not equal to $\log(β)$ then we have a finite time algorithm to determine whether or not $\mathrm{dim}_\mathrm{H} (ν_β)=1$.

math.CA↗
arXiv2017

Lyapunov exponents for products of matrices

Let ${\bf M}=(M_1,\ldots, M_k)$ be a tuple of real $d\times d$ matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether ${\bf M}$ possesses the following property: there exist two constants $λ\in {\Bbb R}$ and $C>0$ such that for any $n\in {\Bbb N}$ and any $i_1, \ldots, i_n \in \{1,\ldots, k\}$, either $M_{i_1} \cdots M_{i_n}={\bf 0}$ or $C^{-1} e^{λn} \leq \| M_{i_1} \cdots M_{i_n} \| \leq C e^{λn}$, where $\|\cdot\|$ is a matrix norm. The proof is based on symbolic dynamics and the thermodynamic formalism for matrix products. As applications, we are able to check the absolute continuity of a class of overlapping self-similar measures on ${\Bbb R}$, the absolute continuity of certain self-affine measures in ${\Bbb R}^d$ and the dimensional regularity of a class of sofic affine-invariant sets in the plane.

math.DS↗
arXiv2016

Affine embeddings of Cantor sets and dimension of $αβ$-sets

Let $E, F\subset {\Bbb R}^d$ be two self-similar sets, and suppose that $F$ can be affinely embedded into $E$. Under the assumption that $E$ is dust-like and has a small Hausdorff dimension, we prove the logarithmic commensurability between the contraction ratios of $E$ and $F$. This gives a partial affirmative answer to Conjecture 1.2 in \cite{FHR14}. The proof is based on our study of the box-counting dimension of a class of multi-rotation invariant sets on the unit circle, including the $αβ$-sets initially studied by Engelking and Katznelson.

math.CA↗
arXiv2016

Projections of planar Mandelbrot measures

Let $μ$ be a planar Mandelbrot measure and $π_*μ$ its orthogonal projection on one of the main axes. We study the thermodynamic and geometric properties of $π_*μ$. We first show that $π_*μ$ is exactly dimensional, with $\dim(π_*μ)=\min(\dim(μ),\dim(ν))$, where~$ν$ is the Bernoulli product measure obtained as the expectation of $π_*μ$. We also prove that $π_*μ$ is absolutely continuous with respect to $ν$ if and only if $\dim(μ)>\dim(ν)$, and find sufficient conditions for the equivalence of these measures. Our results provides a new proof of Dekking-Grimmett-Falconer formula for the Hausdorff and box dimension of the topological support of $π_*μ$, as well as a new variational interpretation. We obtain the free energy function $τ_{π_*μ}$ of $π_*μ$ on a wide subinterval $[0,q_c)$ of $\mathbb{R}_+$. For $q\in[0,1]$, it is given by a variational formula which sometimes yields phase transitions of order larger than~1. For $q>1$, it is given by $\min(τ_ν,τ_μ)$, which can exhibit first order phase transitions. This is in contrast with the analyticity of $τ_μ$ over $[0,q_c)$. Also, we prove the validity of the multifractal formalism for $π_*μ$ at each $α\in (τ_{π_*μ}'(q_c-),τ_{π_*μ}'(0+)]$.

math.PR↗
arXiv2015

Dimensions of random covering sets in Riemann manifolds

Let ${\pmb M}$, ${\pmb N}$ and ${\pmb K}$ be $d$-dimensional Riemann manifolds. Assume that ${\bf A}:=(A_n)_{n\in{\Bbb N}}$ is a sequence of Lebesgue measurable subsets of ${\pmb M}$ satisfying a necessary density condition and ${\bf x}:=(x_n)_{n\in {\Bbb N}}$ is a sequence of independent random variables which are distributed on ${\pmb K}$ according to a measure which is not purely singular with respect to the Riemann volume. We give a formula for the almost sure value of the Hausdorff dimension of random covering sets ${\bf E}({\bf x},{\bf A}):=\limsup_{n\to\infty}A_n(x_n)\subset {\pmb N}$. Here $A_n(x_n)$ is a diffeomorphic image of $A_n$ depending on $x_n$. We also verify that the packing dimensions of ${\bf E}({\bf x},{\bf A})$ equal $d$ almost surely.

math.CA↗