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.contributor.department | Mathematics | 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.date.accessioned | 2015-02-05T06:33:33Z | |
| dc.date.available | 2015-02-05T06:33:33Z | |
| dc.date.issued | 2014 | en_US |
| 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.identifier | https://doi.org/10.13016/M2B02H | |
| dc.identifier.uri | http://hdl.handle.net/1903/16073 | |
| dc.language.iso | en | 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 |
| dc.title | Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps | en_US |
| dc.type | Dissertation | en_US |
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