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Search Results: 1 - 10 of 237686 matches for " De-Jun Feng "
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Weighted equilibrium states for factor maps between subshifts
De-Jun Feng
Mathematics , 2009,
Abstract: Let $\pi:X\to Y$ be a factor map, where $(X,\sigma_X)$ and $(Y,\sigma_Y)$ 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 (H\"{o}lder continuity, for instance). We show that there is a unique shift invariant measure $\mu$ on $X$ that maximizes $\mu(f)+a_1h_\mu(\sigma_X)+ a_2h_{\mu\circ \pi^{-1}}(\sigma_Y)$. In particular, taking $f\equiv 0$ we see that there is a unique invariant measure $\mu$ on $X$ that maximizes the weighted entropy $a_1h_\mu(\sigma_X)+ a_2h_{\mu\circ \pi^{-1}}(\sigma_Y)$. 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.
Multifractal analysis of Bernoulli convolutions associated with Salem numbers
De-Jun Feng
Mathematics , 2011,
Abstract: We consider the multifractal structure of the Bernoulli convolution $\nu_{\lambda}$, where $\lambda^{-1}$ is a Salem number in $(1,2)$. Let $\tau(q)$ denote the $L^q$ spectrum of $\nu_\lambda$. We show that if $\alpha \in [\tau'(+\infty), \tau'(0+)]$, then the level set $$E(\alpha):={x\in \R:\; \lim_{r\to 0}\frac{\log \nu_\lambda([x-r, x+r])}{\log r}=\alpha}$$ is non-empty and $\dim_HE(\alpha)=\tau^*(\alpha)$, where $\tau^*$ denotes the Legendre transform of $\tau$. This result extends to all self-conformal measures satisfying the asymptotically weak separation condition. We point out that the interval $[\tau'(+\infty), \tau'(0+)]$ is not a singleton when $\lambda^{-1}$ 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.
On the topology of polynomials with bounded integer coefficients
De-Jun Feng
Mathematics , 2011,
Abstract: For a real number $q>1$ and a positive integer $m$, let $Y_m(q):={\sum_{i=0}^n\epsilon_i q^i:\; \epsilon_i\in \{0, \pm 1,..., \pm m\}, n=0, 1,...}.$ In this paper, we show that $Y_m(q)$ is dense in ${\Bbb R}$ if and only if $q
Study of Temperature Dependence of Holographic Grating Fabricated in Photopolymer

FENG De-jun,

光子学报 , 2007,
Abstract: 在Polyacrylic Acid(PPA)光聚合物中用双光束相干的方法成功地写入了全息光栅,用He-Ne激光器作为探测光对光栅的写入过程进行了实时监测.分两种情况重点研究了温度对光栅成栅过程的影响,一是在不同的温度下写入光栅;二是在室温下写入光栅,待光栅强度达到最高时再改变温度来研究温度对光栅的影响.实验中温度的变化范围为25℃~100℃.
Equilibrium states of the pressure function for products of matrices
De-Jun Feng,Antti Kaenmaki
Mathematics , 2010,
Abstract: 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.
Variational principles for topological entropies of subsets
De-Jun Feng,Wen Huang
Mathematics , 2010,
Abstract: Let $(X,T)$ be a topological dynamical system. We define the measure-theoretical lower and upper entropies $\underline{h}_\mu(T)$, $\bar{h}_\mu(T)$ for any $\mu\in M(X)$, 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 $$\htop^B(T, K)= \sup \{\underline{h}_\mu(T): \mu\in M(X),\; \mu(K)=1\}, $$ $$\htop^P(T, K)= \sup \{\bar{h}_\mu(T): \mu\in M(X),\; \mu(K)=1\}. $$ 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 $X$.
Multifractal formalism for almost all self-affine measures
Julien Barral,De-Jun Feng
Mathematics , 2011,
Abstract: 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.
Non-conformal repellers and the continuity of pressure for matrix cocycles
De-Jun Feng,Pablo Shmerkin
Mathematics , 2013, DOI: 10.1007/s00039-014-0274-7
Abstract: 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.
Growth rate for beta-expansions
De-Jun Feng,Nikita Sidorov
Mathematics , 2009,
Abstract: Let $\beta>1$ and let $m>\be$ be an integer. Each $x\in I_\be:=[0,\frac{m-1}{\beta-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty \epsilon_k\beta^{-k}, \] where $\epsilon_k\in\{0,1,...,m-1\}$ for all $k$ (a $\beta$-expansion of $x$). It is known that a.e. $x\in I_\beta$ has a continuum of distinct $\beta$-expansions. In this paper we prove that if $\beta$ 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 $\beta$. When $\beta<\frac{1+\sqrt5}2$, we show that the set of $\beta$-expansions grows exponentially for every internal $x$.
Variational principle for weighted topological pressure
De-Jun Feng,Wen Huang
Mathematics , 2014,
Abstract: Let $\pi:X\to Y$ 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: $$ P^{\bf a}(X, f)=\sup\left\{a_1h_\mu(T)+a_2h_{\mu\circ\pi^{-1}}(S)+\int f \;d\mu\right\}, $$ 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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