UMD Theses and Dissertations

Permanent URI for this collectionhttp://hdl.handle.net/1903/3

New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a given thesis/dissertation in DRUM.

More information is available at Theses and Dissertations at University of Maryland Libraries.

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2014 - 20202

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    Dynamics of Some Fermi Acceleration Models
    (2020) Zhou, Jing; Dolgopyat, Dmitry; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    In the thesis we describe the dynamics of two variations of the Fermi acceleration models. The first model consists of a rectangular billiard with two periodically vertically oscillating slits. A point particle bounces elastically against the billiard table and the slits. We assume that the horizontal motion of the particle is in resonance with those of the slits. In this case, we have found a mechanism of trapping regions which provides the exponential acceleration for almost all initial conditions with sufficiently high initial energy. Under an additional hyperbolicity assumption on the parameters of the system, we estimate the waiting time after which most high-energy orbits start to gain energy exponentially fast. The second model depicts a point particle bouncing elastically against a periodically oscillating platform in a gravity field. We assume that the platform motion is piecewise smooth with one singularity. If the second derivative of the platform motion behaves well, i.e. it is either always positive or always less than the negative of the gravitational constant, then the escaping orbits constitute a null set and the system is recurrent. However, under these assumptions, escaping orbits coexist with bounded orbits at arbitrarily high energy levels.
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    Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps
    (2014) Contreras, Fabian Elias; Dolgopyat, Dmitry; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    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.