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Lior Fishman

  1. Schmidt's game, fractals, and orbits of toral endomorphisms.

    Втр, 01/05/2010 - 16:01 — SomNambul
    Authors: Lior Fishman, Ryan Broderick, Dmitry Kleinbock
    Subjects: Dynamical Systems
    Abstract

    Given an integer nonsingular $n\times bn$ matrix $M$ and a point $y \in
    \mathbb{R}^n/\mathbb{Z}^n$, consider the set $\tilde E(M,y)$ of vectors $x\in
    \mathbb{R}^n$ such that $y$ is not a limit point of the sequence $\{M^k x \mod
    \Z^n: k\in\N\}$. S.G. Dani showed in 1988 that whenever $M$ is semisimple and
    $y \in \mathbb{Q}^n/\mathbb{Z}^n$, the set $\tilde E(M,y)$ is winning in the
    sense of W.

  2. Schmidt's game, fractals, and numbers normal to no base.

    Чт, 09/24/2009 - 12:35 — SomNambul
    Authors: Lior Fishman, Yann Bugeaud, Ryan Broderick, Dmitry Kleinbock, Barak Weiss
    Subjects: Dynamical Systems
    Abstract

    Given b > 1 and y \in \mathbb{R}/\mathbb{Z}$, we consider the set of $x\in
    \mathbb{R}$ such that $y$ is not a limit point of the sequence $\{b^n x \bmod
    1: n\in\N\}$. Such sets are known to have full Hausdorff dimension, and in many
    cases have been shown to have a stronger property of being winning in the sense
    of Schmidt. In this paper, by utilizing Schmidt games, we prove that these sets
    and their bi-Lipschitz images must intersect with `sufficiently regular'
    fractals $K\subset \mathbb{R}$ (that is, supporting measures $\mu$ satisfying
    certain decay conditions).

  3. Schmidt's game, fractals, and numbers normal to no base.

    Чт, 09/24/2009 - 12:35 — SomNambul
    Authors: Lior Fishman, Yann Bugeaud, Ryan Broderick, Dmitry Kleinbock, Barak Weiss
    Subjects: Dynamical Systems
    Abstract

    Given b > 1 and y \in \mathbb{R}/\mathbb{Z}$, we consider the set of $x\in
    \mathbb{R}$ such that $y$ is not a limit point of the sequence $\{b^n x \bmod
    1: n\in\N\}$. Such sets are known to have full Hausdorff dimension, and in many
    cases have been shown to have a stronger property of being winning in the sense
    of Schmidt. In this paper, by utilizing Schmidt games, we prove that these sets
    and their bi-Lipschitz images must intersect with `sufficiently regular'
    fractals $K\subset \mathbb{R}$ (that is, supporting measures $\mu$ satisfying
    certain decay conditions).

  4. Diophantine approximations on fractals.

    Втр, 08/18/2009 - 11:53 — SomNambul
    Authors: Manfred Einsiedler, Lior Fishman, Uri Shapira
    Subjects: Dynamical Systems
    Abstract

    We exploit dynamical properties of diagonal actions to derive results in
    Diophantine approximations. In particular, we prove that the continued fraction
    expansion of almost any point on the middle third Cantor set (with respect to
    the natural measure) contains all finite patterns (hence is well approximable).
    Similarly, we show that for a variety of fractals in [0,1]^2, possessing some
    symmetry, almost any point is not Dirichlet improvable (hence is well
    approximable) and has property C (after Cassels). We then settle by similar
    methods a conjecture of M.

Math-arch

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