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Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps

dc.contributor.advisorDolgopyat, Dmitryen_US
dc.contributor.authorContreras, Fabian Eliasen_US
dc.description.abstractThis 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.titleRegularity of absolutely continuous invariant measures for piecewise expanding unimodal mapsen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.subject.pqcontrolledTheoretical mathematicsen_US
dc.subject.pquncontrolleddynamical systemsen_US
dc.subject.pquncontrolledergodic theoryen_US
dc.subject.pquncontrolledpiecewise expanding unimodalen_US
dc.subject.pquncontrolledTaylor seriesen_US
dc.subject.pquncontrolledWhitney senseen_US

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