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De-Jun Feng

Publications and source records attributed to De-Jun Feng.

36 records · Page 2Linked to original sources
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↗
arXiv2015

On the topology of polynomials with bounded integer coefficients

For a real number $q>1$ and a positive integer $m$, let $Y_m(q):={\sum_{i=0}^nε_i q^i:\; ε_i\in \{0, \pm 1,..., \pm m\}, n=0, 1,...}.$ In this paper, we show that $Y_m(q)$ is dense in ${\Bbb R}$ if and only if $q<m+1$ and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik.

math.NT↗
arXiv2014

Variational principle for weighted topological pressure

Let $π:X\to Y$ be a factor map, where $(X,T)$ and $(Y,S)$ are topological dynamical systems. Let ${\bf a}=(a_1,a_2)\in {\Bbb R}^2$ with $a_1>0$ and $a_2\geq 0$, and $f\in C(X)$. The ${\bf a}$-weighted topological pressure of $f$, denoted by $P^{\bf a}(X, f)$, is defined by resembling the Hausdorff dimension of subsets of self-affine carpets. We prove the following variational principle: $$ P^{\bf a}(X, f)=\sup\left\{a_1h_μ(T)+a_2h_{μ\circπ^{-1}}(S)+\int f \;dμ\right\}, $$ where the supremum is taken over the $T$-invariant measures on $X$. It not only generalizes the variational principle of classical topological pressure, but also provides a topological extension of dimension theory of invariant sets and measures on the torus under affine diagonal endomorphisms. A higher dimensional version of the result is also established.

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arXiv2014

Self-similar subsets of the Cantor set

In this paper, we study the following question raised by Mattila in 1998: what are the self-similar subsets of the middle-third Cantor set $\C$? We give criteria for a complete classification of all such subsets. We show that for any self-similar subset $\F$ of $\C$ containing more than one point every linear generating IFS of $\F$ must consist of similitudes with contraction ratios $\pm 3^{-n}$, $n\in \N$. In particular, a simple criterion is formulated to characterize self-similar subsets of $\C$ with equal contraction ratio in modulus.

math.DS↗
arXiv2014

Affine embeddings and intersections of Cantor sets

Let $E, F\subset \R^d$ be two self-similar sets. Under mild conditions, we show that $F$ can be $C^1$-embedded into $E$ if and only if it can be affinely embedded into $E$; furthermore if $F$ can not be affinely embedded into $E$, then the Hausdorff dimension of the intersection $E\cap f(F)$ is strictly less than that of $F$ for any $C^1$-diffeomorphism $f$ on $\R^d$. Under certain circumstances, we prove the logarithmic commensurability between the contraction ratios of $E$ and $F$ if $F$ can be affinely embedded into $E$. As an application, we show that $\dim_HE\cap f(F)<\min\{\dim_HE, \dim_HF\}$ when $E$ is any Cantor-$p$ set and $F$ any Cantor-$q$ set, where $p,q\geq 2$ are two integers with $\log p/\log q\not \in \Q$. This is related to a conjecture of Furtenberg about the intersections of Cantor sets.

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arXiv2014

Non-conformal repellers and the continuity of pressure for matrix cocycles

The pressure function $P(A, s)$ plays a fundamental role in the calculation of the dimension of "typical" self-affine sets, where $A=(A_1,\ldots, A_k)$ is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on $A$. As a consequence, we show that the dimension of "typical" self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated by the norm of matrix products or generalized singular value functions for matrix cocycles, and obtain applications on the continuity of equilibrium measures and the Lyapunov spectrum of matrix cocycles.

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arXiv2012

Multifractal formalism for almost all self-affine measures

We conduct the multifractal analysis of self-affine measures for "almost all" family of affine maps. Besides partially extending Falconer's formula of $L^q$-spectrum outside the range $1< q\leq 2$, the multifractal formalism is also partially verified.

math.CA↗
arXiv2012

Local conical dimensions for measures

We study the decay of $μ(B(x,r)\cap C)/μ(B(x,r))$ as $r\downarrow 0$ for different kinds of measures $μ$ on $\R^n$ and various cones $C$ around $x$. As an application, we provide sufficient conditions implying that the local dimensions can be calculated via cones almost everywhere.

math.CA↗
arXiv2011

Multifractal analysis of Bernoulli convolutions associated with Salem numbers

We consider the multifractal structure of the Bernoulli convolution $ν_λ$, where $λ^{-1}$ is a Salem number in $(1,2)$. Let $τ(q)$ denote the $L^q$ spectrum of $ν_λ$. We show that if $α\in [τ'(+\infty), τ'(0+)]$, then the level set $$E(α):={x\in \R:\; \lim_{r\to 0}\frac{\log ν_λ([x-r, x+r])}{\log r}=α}$$ is non-empty and $\dim_HE(α)=τ^*(α)$, where $τ^*$ denotes the Legendre transform of $τ$. This result extends to all self-conformal measures satisfying the asymptotically weak separation condition. We point out that the interval $[τ'(+\infty), τ'(0+)]$ is not a singleton when $λ^{-1}$ is the largest real root of the polynomial $x^{n}-x^{n-1}-... -x+1$, $n\geq 4$. An example is constructed to show that absolutely continuous self-similar measures may also have rich multifractal structures.

math.CA↗
arXiv2010

Variational principles for topological entropies of subsets

Let $(X,T)$ be a topological dynamical system. We define the measure-theoretical lower and upper entropies $\underline{h}_μ(T)$, $\bar{h}_μ(T)$ for any $μ\in M(X)$, where $M(X)$ denotes the collection of all Borel probability measures on $X$. For any non-empty compact subset $K$ of $X$, we show that $$\htop^B(T, K)= \sup \{\underline{h}_μ(T): μ\in M(X),\; μ(K)=1\}, $$ $$\htop^P(T, K)= \sup \{\bar{h}_μ(T): μ\in M(X),\; μ(K)=1\}. $$ where $\htop^B(T, K)$ denotes Bowen's topological entropy of $K$, and $\htop^P(T, K)$ the packing topological entropy of $K$. Furthermore, when $\htop(T)<\infty$, the first equality remains valid when $K$ is replaced by an arbitrarily analytic subset of $X$. The second equality always extends to any analytic subset of $X$.

math.DS↗
arXiv2010

Equilibrium states of the pressure function for products of matrices

Let $\{M_i\}_{i=1}^\ell$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \N$, there exists $i_1... i_n\in \{1,..., \ell\}^n$ such that $M_{i_1}... M_{i_n}\neq {\bf 0}$. Let $P \colon (0,\infty)\to \R$ be the pressure function of $\{M_i\}_{i=1}^\ell$. We show that for each $q>0$, there are at most $d$ ergodic $q$-equilibrium states of $P$, and each of them satisfies certain Gibbs property.

math.DS↗
arXiv2010

Dimension theory of iterated function systems

Let $\{S_i\}_{i=1}^\ell$ be an iterated function system (IFS) on $\R^d$ with attractor $K$. Let $(Σ,σ)$ denote the one-sided full shift over the alphabet $\{1,..., \ell\}$. We define the projection entropy function $h_π$ on the space of invariant measures on $Σ$ associated with the coding map $π: Σ\to K$, and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on $K$. We show that for any conformal IFS (resp., the direct product of finitely many conformal IFS), without any separation condition, the projection of an ergodic measure under $π$ is always exactly dimensional and, its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (resp., the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFS, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures.

math.DS↗
arXiv2010

Lyapunov spectrum of asymptotically sub-additive potentials

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measure-theoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the existence of unique equilibrium measures or the differentiability of pressure functions. Some examples are constructed to illustrate the irregularity and the complexity of multifractal behaviors in the sub-additive case and in the case that the entropy map that is not upper-semi continuous.

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arXiv2009

Growth rate for beta-expansions

Let $β>1$ and let $m>\be$ be an integer. Each $x\in I_\be:=[0,\frac{m-1}{β-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty ε_kβ^{-k}, \] where $ε_k\in\{0,1,...,m-1\}$ for all $k$ (a $β$-expansion of $x$). It is known that a.e. $x\in I_β$ has a continuum of distinct $β$-expansions. In this paper we prove that if $β$ is a Pisot number, then for a.e. $x$ this continuum has one and the same growth rate. We also link this rate to the Lebesgue-generic local dimension for the Bernoulli convolution parametrized by $β$. When $β<\frac{1+\sqrt5}2$, we show that the set of $β$-expansions grows exponentially for every internal $x$.

math.NT↗
arXiv2009

Weighted thermodynamic formalism and applications

Let $(X,T)$ and $(Y,S)$ be two subshifts so that $Y$ is a factor of $X$. For any asymptotically sub-additive potential $Φ$ on $X$ and $\ba=(a,b)\in\R^2$ with $a>0$, $b\geq 0$, we introduce the notions of $\ba$-weighted topological pressure and $\ba$-weighted equilibrium state of $Φ$. We setup the weighted variational principle. In the case that $X, Y$ are full shifts with one-block factor map, we prove the uniqueness and Gibbs property of $\ba$-weighted equilibrium states for almost additive potentials having the bounded distortion properties. Extensions are given to the higher dimensional weighted thermodynamic formalism. As an application, we conduct the multifractal analysis for a new type of level sets associated with Birkhoff averages, as well as for weak Gibbs measures associated with asymptotically additive potentials on self-affine symbolic spaces.

math.DS↗
arXiv2009

Weighted equilibrium states for factor maps between subshifts

Let $π:X\to Y$ be a factor map, where $(X,σ_X)$ and $(Y,σ_Y)$ are subshifts over finite alphabets. Assume that $X$ satisfies weak specification. Let $\ba=(a_1,a_2)\in \R^2$ with $a_1>0$ and $a_2\geq 0$. Let $f$ be a continuous function on $X$ with sufficient regularity (Hölder continuity, for instance). We show that there is a unique shift invariant measure $μ$ on $X$ that maximizes $μ(f)+a_1h_μ(σ_X)+ a_2h_{μ\circ π^{-1}}(σ_Y)$. In particular, taking $f\equiv 0$ we see that there is a unique invariant measure $μ$ on $X$ that maximizes the weighted entropy $a_1h_μ(σ_X)+ a_2h_{μ\circ π^{-1}}(σ_Y)$. This answers an open question raised by Gatzouras and Peres in \cite{GaPe96}. An extension is also given to high dimensional cases. As an application, we show the uniqueness of invariant measures with full Hausdorff dimension for certain affine invariant sets on the $k$-torus under a diagonal endomorphism.

math.DS↗
arXiv2002

The Pressure Function for Products of Non-negative Matrices

Let $(Σ_A, σ)$ be a subshift of finite type and let $M(x)$ be a continuous function on $Σ_A$ taking values in the set of non-negative matrices. We extend the classical scalar pressure function to this new setting and prove the existence of the Gibbs measure and the differentiability of the pressure function. We are especially interested on the case where $M(x)$ takes finite values $M_1, ..., M_m$. The pressure function reduces to $P(q):=\lim_{n\to \infty}\frac{1}{n} \log \sum_{J \in \sum_{A, n}} \|M_J\|^q$. The expression is important when we consider the multifractal formalism for certain iterated function systems with overlaps.

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