Mathematics
Permanent URI for this communityhttp://hdl.handle.net/1903/2261
Browse
6 results
Search Results
Item Absolutely Continuous Spectrum for Parabolic Flows/Maps(2016) Simonelli, Lucia Dora; Forni, Giovanni; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This work is devoted to creating an abstract framework for the study of certain spectral properties of parabolic systems. Specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these general conditions to derive results for spectral properties of time-changes of unipotent flows on homogeneous spaces of semisimple groups regarding absolutely continuous spectrum as well as maximal spectral type; the time-changes of the horocycle flow are special cases of this general category of flows. In addition we use the general conditions to derive spectral results for twisted horocycle flows and to rederive spectral results for skew products over translations and Furstenberg transformations.Item Progress Toward Classifying Teichmueller Disks with Completely Degenerate Kontsevich-Zorich Spectrum(2012) Aulicino, David; Forni, Giovanni; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We present results toward resolving a question posed by Eskin-Kontsevich-Zorich and Forni-Matheus-Zorich. They asked for a classification of all $\text{SL}_2(\mathbb{R})$-invariant ergodic probability measures with completely degenerate Kontsevich - Zorich spectrum. Let $\mathcal{D}_g(1)$ be the subset of the moduli space of Abelian differentials $mathcal{M}_g$ whose elements have period matrix derivative of rank one. There is an $\text{SL}_2(\mathbb{R})$-invariant ergodic probability measure $nu$ with completely degenerate Kontsevich-Zorich spectrum, i.e. $lambda_1 = 1 > lambda_2 = cdots = lambda_g = 0$, if and only if $nu$ has support contained in $\mathcal{D}_g(1)$. We approach this problem by studying Teichm"uller disks contained in $\mathcal{D}_g(1)$. We show that if $(X,omega)$ generates a Teichm"uller disk in $\mathcal{D}_g(1)$, then $(X,omega)$ is completely periodic. Furthermore, we show that there are no Teichm"uller disks in $\mathcal{D}_g(1)$, for $g = 2$, and the known example of a Teichm"uller disk in $mathcal{D}_3(1)$ is the only one. Finally, we present an idea that might be able to fully resolve the problem.Item The Cohomological Equation for Horocycle Maps and Quantitative Equidistribution(2011) Tanis, James Holloway; Forni, Giovanni; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)There are infinitely many distributional obstructions to the existence of smooth solutions for the cohomological equation u o φ1 - u = f in each irreducible component of L2(Γ\PSL(2,R)), where φ1 is the time-one map of the horocycle flow. We study the regularity of these obstructions, determine which ones also obstruct the existence of L2 solutions and prove a Sobolev estimate of the solution in terms of f. As an application, we estimate the rate of equidistribution of horocycle maps on compact, finite volume manifolds Γ\PSL(2,R)) using an auxiliary result from Flaminio-Forni (2003) and one from Venkatesh (2010) concerning the horocycle flow and the twisted horocycle flow, respectively.Item Cometary Escape in the Restricted Circular Planar Three Body Problem(2011) Galante, Joseph Robert; Kaloshin, Vadim Yu; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The classical principle of least action says that orbits of mechanical systems extremize action; an important subclass are those orbits that minimize action. This principle is utilized along with Aubry-Mather theory to construct regions of instability for a certain three body problem, given by a Hamiltonian system of two degrees of freedom. In principle, these methods can be applied to construct instability regions for a variety of Hamiltonian systems with $2$ degrees of freedom. The Hamiltonian model considered in this thesis describes the dynamics of a Sun-Jupiter-Comet system and under some simplifying assumptions, the existence of instabilities for the orbit of the comet is shown. In particular it is shown that a comet which starts close to an orbit in the shape of an ellipse of eccentricity $e=0.66$ can increase in eccentricity to beyond $e=1$. Furthermore, there exist ejection orbits for the comet. Such orbits are initially well within the range of our solar system. This might give an indication of why most objects rotating around the Sun in our solar system have relatively low eccentricity. Several new theoretical tools are introduced in this thesis as well. The most notable is a checkable sufficient condition to verify that an exact area preserving map is an exact area preserving twist map in a certain coordinate system. This coordinate system is constructed by ``spreading the cumulative twist'' which arises from the long term dynamics of system. Many of the results of the thesis are `computer assisted' and utilize recent advances in rigorous numerical integration. It is through the application of these advances in computing that it has become possible to state deep results for realistic solar systems. This has been the dream of many since humans first observed the stars so long ago.Item Making Forecasts for Chaotic Processes in the Presence of Model Error(2006-02-20) Danforth, Christopher M; Yorke, James A; Kalnay, Eugenia; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Numerical weather forecast errors are generated by model deficiencies and by errors in the initial conditions which interact and grow nonlinearly. With recent progress in data assimilation, the accuracy in the initial conditions has been substantially improved so that accounting for systematic errors associated with model deficiencies has become even more important to ensemble prediction and data assimilation applications. This dissertation describes two new methods for reducing the effect of model error in forecasts. The first method is inspired by Leith (1978) who proposed a statistical method to account for model bias and systematic errors linearly dependent on the flow anomalies. DelSole and Hou (1999) showed this method to be successful when applied to a very low order quasi-geostrophic model simulation with artificial "model errors." However, Leith's method is computationally prohibitive for high-resolution operational models. The purpose of the present study is to explore the feasibility of estimating and correcting systematic model errors using a simple and efficient procedure that could be applied operationally, and to compare the impact of correcting the model integration with statistical corrections performed a posteriori. The second method is inspired by the dynamical systems theory of shadowing. Making a prediction for a chaotic physical process involves specifying the probability associated with each possible outcome. Ensembles of solutions are frequently used to estimate this probability distribution. However, for a typical chaotic physical system H and model L of that system, no solution of L remains close to H for all time. We propose an alternative and show how to "inflate" or systematically perturb the ensemble of solutions of L so that some ensemble member remains close to H for orders of magnitude longer than unperturbed solutions of L. This is true even when the perturbations are significantly smaller than the model error.Item Finding Optimal Orbits of Chaotic Systems(2005-12-05) Grant, Angela Elyse; Hunt, Brian R; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Chaotic dynamical systems can exhibit a wide variety of motions, including periodic orbits of arbitrarily large period. We consider the question of which motion is optimal, in the sense that it maximizes the average over time of some given scalar ``performance function." Past work indicates that optimal motions tend to be periodic orbits with low period, but does not describe, beyond a brute force approach, how to determine which orbit is optimal in a particular scenario. For one-dimensional expanding maps and higher dimensional hyperbolic systems, we have found constructive methods for calculating the optimal average and corresponding periodic orbit, and by carrying them out on a computer have found them to work quite well in practice.