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    Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps

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    Contreras_umd_0117E_15654.pdf (269.3Kb)
    No. of downloads: 308

    Date
    2014
    Author
    Contreras, Fabian Elias
    Advisor
    Dolgopyat, Dmitry
    DRUM DOI
    https://doi.org/10.13016/M2B02H
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    Abstract
    This dissertation consists of two parts. In the first part, we consider a piecewise expanding unimodal map (PEUM) $f:[0,1] \to [0,1]$ with $\mu=\rho dx$ the (unique) SRB measure associated to it and we show that $\rho$ has a Taylor expansion in the Whitney sense. Moreover, we prove that the set of points where $\rho$ is not differentiable is uncountable and has Hausdorff dimension equal to zero. In the second part, we consider a family $f_t:[0,1] \to [0,1]$ of PEUMs with $\mu_t$ the correspoding SRB measure and we present a new proof of \cite{BS1} when considering the observables in $C^1[0,1]$ . That is, $\Gamma(t)=\int \phi d\mu_t$ is differentiable at $t=0$, with $\phi \in C^1[0,1]$, when assuming $J(c)=\sum_{k=0}^{\infty} \frac{v(f^k(c))}{Df^k(f(c))}$ is zero. Furthermore, we show that in fact $\Gamma(t)$ is never differentiable when $J(c)$ is not zero and we give the exact modulus of continuity.
    URI
    http://hdl.handle.net/1903/16073
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    DRUM is brought to you by the University of Maryland Libraries
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