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De-Jun Feng

  1. Dimension theory of iterated function systems.

    Thu, 02/11/2010 - 16:01 — SomNambul
    Authors: De-Jun Feng, Huyi Hu
    Subjects: Dynamical Systems
    Abstract

    Let $\{S_i\}_{i=1}^\ell$ be an iterated function system (IFS) on $\R^d$ with
    attractor $K$. Let $(\Sigma,\sigma)$ denote the one-sided full shift over the
    alphabet $\{1,..., \ell\}$. We define the projection entropy function $h_\pi$
    on the space of invariant measures on $\Sigma$ associated with the coding map
    $\pi: \Sigma\to K$, 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 $K$.

  2. Weighted thermodynamic formalism and applications.

    Thu, 09/24/2009 - 12:35 — SomNambul
    Authors: De-Jun Feng, Julien Barral
    Subjects: Dynamical Systems
    Abstract

    Let $(X,T)$ and $(Y,S)$ be two subshifts so that $Y$ is a factor of $X$. For
    any asymptotically sub-additive potential $\Phi$ 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 $\Phi$. 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.

  3. Weighted equilibrium states for factor maps between subshifts.

    Thu, 09/24/2009 - 12:35 — SomNambul
    Authors: De-Jun Feng
    Subjects: Dynamical Systems
    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 $\mu(f)+a_1h_\mu(\sigma_X)+ a_2h_{\mu\circ
    \pi^{-1}}(\sigma_Y)$.

  4. Growth rate for beta-expansions.

    Tue, 08/18/2009 - 11:53 — SomNambul
    Authors: De-Jun Feng, Nikita Sidorov
    Subjects: Number Theory
    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
    $\beta$.

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