# Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps

 dc.contributor.advisor Dolgopyat, Dmitry en_US dc.contributor.author Contreras, Fabian Elias en_US dc.date.accessioned 2015-02-05T06:33:33Z dc.date.available 2015-02-05T06:33:33Z dc.date.issued 2014 en_US dc.identifier https://doi.org/10.13016/M2B02H dc.identifier.uri http://hdl.handle.net/1903/16073 dc.description.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. en_US dc.language.iso en en_US dc.title Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps en_US dc.type Dissertation en_US dc.contributor.publisher Digital Repository at the University of Maryland en_US dc.contributor.publisher University of Maryland (College Park, Md.) en_US dc.contributor.department Mathematics en_US dc.subject.pqcontrolled Theoretical mathematics en_US dc.subject.pquncontrolled dynamical systems en_US dc.subject.pquncontrolled ergodic theory en_US dc.subject.pquncontrolled invariant en_US dc.subject.pquncontrolled piecewise expanding unimodal en_US dc.subject.pquncontrolled Taylor series en_US dc.subject.pquncontrolled Whitney sense en_US
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