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1.
Hughes, David Rhodri.
Entropy of the Lorentz Gas and Other Results.
Degree: 2019, Penn State University
URL: https://submit-etda.libraries.psu.edu/catalog/16864drh272 
► In Chapter 1 we provide relevant definitions and background material on Dynamical Systems and Chaotic Billiards. In Chapter 2 we prove that the topological entropy…
(more)
▼ In Chapter 1 we provide relevant definitions and background material on Dynamical Systems and Chaotic Billiards.
In Chapter 2 we prove that the topological entropy of the Lorentz gas billiard can take any real positive value if the side lengths of the tables are changed (while the area is held fixed). On the other hand, the metric entropy can take any value between 0 and some constant bound M as the side lengths change. We also show that he ratio of the two entropies can take any value in the interval (0,\frac{1}{2}). It is also shown that both continuously depend on the radius of the circular scatterer.
In Chapter 3 we give a new proof of the Gauss-Bonnet Theorem using the inclusion-exclusion formula.
In Chapter 4 we show that the ratio of the metric entropy to the topological entropy of the geodesic flow of a surface of negative curvature can be made to take any value in (0,1) by taking a time change of the flow.
Finally, in Chapter 5 we have a collection of results on the asymptotic vanishing of the topological entropy of the Bunimovich stadium billiard as the table gets longer and thinner (again, volume is held constant), and that any plane curve homeomorphic to the circle can be recovered from its incidence data with a family of lattices.
Advisors/Committee Members: Dmitri Yu Burago, Dissertation Advisor/Co-Advisor, Mark Levi, Committee Chair/Co-Chair, Federico Juan Rodriguez Hertz, Committee Member, Mark Levi, Committee Member, Martin Bojowald, Outside Member.
Subjects/Keywords: Dynamical Systems; Blliards; Entropy
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APA (6th Edition):
Hughes, D. R. (2019). Entropy of the Lorentz Gas and Other Results. (Thesis). Penn State University. Retrieved from https://submit-etda.libraries.psu.edu/catalog/16864drh272
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Hughes, David Rhodri. “Entropy of the Lorentz Gas and Other Results.” 2019. Thesis, Penn State University. Accessed March 03, 2021.
https://submit-etda.libraries.psu.edu/catalog/16864drh272.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Hughes, David Rhodri. “Entropy of the Lorentz Gas and Other Results.” 2019. Web. 03 Mar 2021.
Vancouver:
Hughes DR. Entropy of the Lorentz Gas and Other Results. [Internet] [Thesis]. Penn State University; 2019. [cited 2021 Mar 03].
Available from: https://submit-etda.libraries.psu.edu/catalog/16864drh272.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Hughes DR. Entropy of the Lorentz Gas and Other Results. [Thesis]. Penn State University; 2019. Available from: https://submit-etda.libraries.psu.edu/catalog/16864drh272
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Penn State University
2.
Wei, Daren.
Slow entropy, Kakutani equivalence and parabolic flows.
Degree: 2020, Penn State University
URL: https://submit-etda.libraries.psu.edu/catalog/17530duw170
► In this dissertation, we use slow entropy and Kakutani equivalence tools to study the parabolic flows. We get the estimates of the slow entropy type…
(more)
▼ In this dissertation, we use slow entropy and Kakutani equivalence tools to study the parabolic flows. We get the estimates of the slow entropy type invariants and Kakutani equivalence invariants for unipotent flows and Kakutani equivalence invariants for products of some special flows. As applications, we make progress on classification of ergodic flows under Kakutani equivalence, get a fully description of standard flows on semi-simple Lie group, obtain a family of non-standard homogeneous flows examples and construct a countably many non-Kakutani equivalent family of natural surface flows.
Advisors/Committee Members: Svetlana Katok, Dissertation Advisor/Co-Advisor, Svetlana Katok, Committee Chair/Co-Chair, Yakov B Pesin, Committee Member, Federico Juan Rodriguez Hertz, Committee Member, Martin Bojowald, Outside Member, Mark Levi, Program Head/Chair.
Subjects/Keywords: Slow entropy; Kakutani equivalence; Parabolic flows; Unipotent flows; Special flows; Standardness
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APA ·
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APA (6th Edition):
Wei, D. (2020). Slow entropy, Kakutani equivalence and parabolic flows. (Thesis). Penn State University. Retrieved from https://submit-etda.libraries.psu.edu/catalog/17530duw170
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Wei, Daren. “Slow entropy, Kakutani equivalence and parabolic flows.” 2020. Thesis, Penn State University. Accessed March 03, 2021.
https://submit-etda.libraries.psu.edu/catalog/17530duw170.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Wei, Daren. “Slow entropy, Kakutani equivalence and parabolic flows.” 2020. Web. 03 Mar 2021.
Vancouver:
Wei D. Slow entropy, Kakutani equivalence and parabolic flows. [Internet] [Thesis]. Penn State University; 2020. [cited 2021 Mar 03].
Available from: https://submit-etda.libraries.psu.edu/catalog/17530duw170.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Wei D. Slow entropy, Kakutani equivalence and parabolic flows. [Thesis]. Penn State University; 2020. Available from: https://submit-etda.libraries.psu.edu/catalog/17530duw170
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Penn State University
3.
Banerjee, Shilpak.
REAL-ANALYTIC ABC METHOD ON THE TORUS.
Degree: 2017, Penn State University
URL: https://submit-etda.libraries.psu.edu/catalog/14089sub232
► Nearly half a century ago, Anosov and Katok invented a scheme to produce examples of volume preserving smooth zero entropy diffeomorphisms satisfying interesting dynamical properties.…
(more)
▼ Nearly half a century ago, Anosov and Katok invented a scheme to produce examples of volume
preserving smooth zero entropy diffeomorphisms satisfying interesting dynamical properties. Their
recipe for creating diffeomorphisms later came to be known as the ‘approximation by conjugation’
method or the ‘Anosov-Katok’ method or simply the ‘AbC’ method. Over the years the AbC
method along with its modifications has become a powerful tool for building diffeomorphisms
with rich dynamical behavior in the zero entropy set up.
These diffeomorphisms are obtained as limits of volume preserving periodic diffeomorphisms
on compact smooth manifolds admitting a non trivial action of the circle. The periodic diffeomorphisms are constructed to satisfy some finite version of the targeted dynamical property.
Unfortunately, when one moves from the world of smooth diffeomorphism to the world of
real-analytic diffeomorphisms, the story becomes much more complicated. In fact, the problem is
largely intractable for abstract real-analytic manifolds, but there is some hope after restricting
attention to specific manifolds like a torus or an odd dimensional sphere.
In this dissertation, we introduce a way to implement the AbC method on a torus and build
real-analytic diffeomorphisms. The key idea behind our construction is the fact that step functions
can be approximated by real-analytic functions. On the other hand one can build transformations
on the torus using these step functions that produce a sliding motion, and we are able to show
that a finite composition of such transformations give us enough flexibility to realize a variety
of finite versions of dynamical properties on a torus. As application of our implementation, we
produce real-analytic versions of several smooth AbC diffeomorphisms.
For example, we obtain non standard real-analytic realizations of certain irrational rotations
of the circle on a torus. Non-standard real-analytic realizations of certain ergodic translations of
a torus are also obtained on another torus.
We note that the work of Fathi and Herman modified the original AbC method and made
it possible to control all orbits. Such modified versions known as topological AbC methods
made it possible to obtain properties like unique ergodicity and minimality, and we are able
to reproduce some of those methods in our set up. In particular we are able to show that
non-standard real-analytic realizations of certain irrational rotations of the circle can be obtained
on the two dimensional torus as uniquely ergodic diffeomorphisms. We also construct examples of
real-analytic minimal diffeomorphisms with a finite number of ergodic invariant measures, each of
which are absolutely continuous with respect to the Lebesgue measure.
Finally, we provide real-analytic realizations of a large class of AbC transformations built by
the abstract AbC scheme on a two dimensional torus. These AbC diffeomorphisms we realize are
isomorphic to the (strongly uniform) circular systems with fast growing parameters invented by
Foreman and Weiss.
Advisors/Committee Members: Anatoly Katok, Dissertation Advisor/Co-Advisor, Anatoly Katok, Committee Chair/Co-Chair, Svetlana Katok, Committee Member, Federico Juan Rodriguez Hertz, Committee Member, Martin Bojowald, Outside Member.
Subjects/Keywords: Dynamical systemsAbC; Ergodic Theory; AbC; Real-Analytic
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APA ·
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APA (6th Edition):
Banerjee, S. (2017). REAL-ANALYTIC ABC METHOD ON THE TORUS. (Thesis). Penn State University. Retrieved from https://submit-etda.libraries.psu.edu/catalog/14089sub232
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Banerjee, Shilpak. “REAL-ANALYTIC ABC METHOD ON THE TORUS.” 2017. Thesis, Penn State University. Accessed March 03, 2021.
https://submit-etda.libraries.psu.edu/catalog/14089sub232.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Banerjee, Shilpak. “REAL-ANALYTIC ABC METHOD ON THE TORUS.” 2017. Web. 03 Mar 2021.
Vancouver:
Banerjee S. REAL-ANALYTIC ABC METHOD ON THE TORUS. [Internet] [Thesis]. Penn State University; 2017. [cited 2021 Mar 03].
Available from: https://submit-etda.libraries.psu.edu/catalog/14089sub232.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Banerjee S. REAL-ANALYTIC ABC METHOD ON THE TORUS. [Thesis]. Penn State University; 2017. Available from: https://submit-etda.libraries.psu.edu/catalog/14089sub232
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Penn State University
4.
Harrison, Michael A.
Skew Flat Fibrations and Totally Convex Immersions.
Degree: 2017, Penn State University
URL: https://submit-etda.libraries.psu.edu/catalog/14140mah5044
► A fibration of \Rn by oriented copies of \Rp is called skew if no two fibers intersect nor contain parallel directions. Conditions on p and…
(more)
▼ A fibration of \R
n by oriented copies of \R
p is called skew if no two fibers intersect nor contain parallel directions. Conditions on p and n for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of \R
3 by skew oriented lines was given by Salvai, in analogue with the classification of oriented great circle fibrations of S
3 by Gluck and Warner. We show that Salvai's classification has a topological variation which generalizes to characterize all continuous fibrations of \R
n by skew oriented copies of \R
p. We show that the space of fibrations of \R
3 by skew oriented lines deformation retracts to the subspace of Hopf fibrations, and therefore has the homotopy type of a pair of disjoint copies of S
2. We discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of \C
n (resp. \Ham
n) by skew oriented copies of \C
p (resp. \Ham
p). \\
An immersion f: M → \R
n is called totally convex if for each pair of distinct points x, y of M, the tangent spaces at f(x) and f(y) do not contain parallel directions. We ask: given a smooth manifold M, what is the minimum dimension n=n(M) such that there exists a totally convex immersion M → \R
n? The corresponding question has been classically studied for immersions, embeddings, and many other maps satisfying distinguished differential conditions. The problem for totally convex immersions is especially difficult; for example, we do not know n(S
2). We find that totally convex immersions are related to the generalized vector field problem, immersions and embeddings of real projective spaces, and the existence of nonsingular symmetric bilinear maps. We provide both upper and lower bounds on the number n. \\
The first chapter, dedicated to skew fibrations, is almost entirely the original work of the author and appeared in similar form in Feb. 2016 in Math. Z. The second chapter, dedicated to totally convex immersions, is similar in spirit to the treatment of totally skew embeddings by Ghomi and Tabachnikov, but the statements are technically new. The results of Chapter 2, along with the future work outlined in the introduction of Chapter 2, will be the subject of a forthcoming article.
Advisors/Committee Members: Sergei Tabachnikov, Dissertation Advisor/Co-Advisor, Sergei Tabachnikov, Committee Chair/Co-Chair, Mark Levi, Committee Member, Federico Juan Rodriguez Hertz, Committee Member, Emily Rolfe Grosholz, Outside Member.
Subjects/Keywords: skew flat fibrations; great sphere fibrations; totally convex immersions; totally skew embeddings; nonsingular bilinear maps; immersions of projective spaces
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APA ·
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CSE |
Export
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Manager
APA (6th Edition):
Harrison, M. A. (2017). Skew Flat Fibrations and Totally Convex Immersions. (Thesis). Penn State University. Retrieved from https://submit-etda.libraries.psu.edu/catalog/14140mah5044
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Harrison, Michael A. “Skew Flat Fibrations and Totally Convex Immersions.” 2017. Thesis, Penn State University. Accessed March 03, 2021.
https://submit-etda.libraries.psu.edu/catalog/14140mah5044.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Harrison, Michael A. “Skew Flat Fibrations and Totally Convex Immersions.” 2017. Web. 03 Mar 2021.
Vancouver:
Harrison MA. Skew Flat Fibrations and Totally Convex Immersions. [Internet] [Thesis]. Penn State University; 2017. [cited 2021 Mar 03].
Available from: https://submit-etda.libraries.psu.edu/catalog/14140mah5044.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Harrison MA. Skew Flat Fibrations and Totally Convex Immersions. [Thesis]. Penn State University; 2017. Available from: https://submit-etda.libraries.psu.edu/catalog/14140mah5044
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Penn State University
5.
Chen, Dong.
On some problems in Lagrangian Dynamics and Finsler Geometry.
Degree: 2017, Penn State University
URL: https://submit-etda.libraries.psu.edu/catalog/14421dxc360
► The purpose of this dissertation is to present several applications of enveloping functions and dual lens maps to geometry and dynamical systems. In Chapter 1…
(more)
▼ The purpose of this dissertation is to present several applications of enveloping functions and dual lens maps to geometry and dynamical systems. In Chapter 1 we have a brief review on basic notions and theory we need to understand the main results. In Chapter 2 we prove that given a point on a Finsler surface, one can always find a neighborhood of the point and isometrically embed this neighborhood into a Finsler torus without conjugate points. The major tool is enveloping functions.
In Chapter 3 we introduce the dual lens map technique developed by Burago and Ivanov. It derives from enveloping functions and symplectic geometry. We then show how this technique is used to perturb the geodesic flows of flat Finsler tori.
In Chapter 4 we show how dual lens map can be used in KAM theory. The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. We show two types of Lagrangian perturbations of the geodesic flow on flat Finsler tori. The perturbations are C^∞ small but the resulting flows has a positive measure of trajectories with positive Lyapunov exponent. The measure of this set is of course extremely small. Still, the flow has positive metric entropy. From this result we get positive metric entropy outside some KAM tori and it gives positive answer to a question asked by Kolmogorov.
Advisors/Committee Members: Dmitri Yu Burago, Dissertation Advisor/Co-Advisor, Dmitri Yu Burago, Committee Chair/Co-Chair, Federico Juan Rodriguez Hertz, Committee Member, Mark Levi, Committee Member, Runze Li, Outside Member.
Subjects/Keywords: KAM theory; Finsler metric; duel lens map Hamiltonian flow; perturbation; metric entropy; duel lens map; Hamiltonian flow; perturbation; metric entropy
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APA ·
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MLA ·
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CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Chen, D. (2017). On some problems in Lagrangian Dynamics and Finsler Geometry. (Thesis). Penn State University. Retrieved from https://submit-etda.libraries.psu.edu/catalog/14421dxc360
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Chen, Dong. “On some problems in Lagrangian Dynamics and Finsler Geometry.” 2017. Thesis, Penn State University. Accessed March 03, 2021.
https://submit-etda.libraries.psu.edu/catalog/14421dxc360.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Chen, Dong. “On some problems in Lagrangian Dynamics and Finsler Geometry.” 2017. Web. 03 Mar 2021.
Vancouver:
Chen D. On some problems in Lagrangian Dynamics and Finsler Geometry. [Internet] [Thesis]. Penn State University; 2017. [cited 2021 Mar 03].
Available from: https://submit-etda.libraries.psu.edu/catalog/14421dxc360.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Chen D. On some problems in Lagrangian Dynamics and Finsler Geometry. [Thesis]. Penn State University; 2017. Available from: https://submit-etda.libraries.psu.edu/catalog/14421dxc360
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Penn State University
6.
Zydney, Adam J.
Boundary maps and their natural extensions associated with Fuchsian and Kleinian groups.
Degree: 2018, Penn State University
URL: https://submit-etda.libraries.psu.edu/catalog/15102ajz5041
► Geodesic flows on surfaces of constant negative curvature are a rich source of examples in ergodic theory, and geodesic flow on the modular surface in…
(more)
▼ Geodesic flows on surfaces of constant negative curvature are a rich source of examples in ergodic theory, and geodesic flow on the modular surface in particular has deep connections to real continued fractions from number theory. This thesis deals with two extensions of this setting: either replacing the modular group PSL(2,Z) with a cocompact torsion-free Fuchsian group, or working with three-dimensional hyperbolic space and relating the boundary maps to continued fractions of complex numbers.
The Fuchsian results are joint with Svetlana Katok and build on results of Katok and Ugarcovici, who studied a family of maps generalizing the Bowen-Series boundary map. When the parameters satisfy the short cycle property, i.e., the forward orbits at each discontinuity point coincide after one step, the natural extension map has a global attractor with finite rectangular structure. In this thesis, we generalize several results of Adler and Flatto, describing a conjugacy between the geometric and arithmetic maps and showing that the attractor parameterizes an associated arithmetic cross-section. This allows us to represent the geodesic flow as a special flow over a symbolic system. In cases where the cycle ends are discontinuity points, the resulting symbolic system is sofic.
In the final chapter, we consider three-dimensional real hyperbolic space, in which the boundary is the Reimann sphere (C with a point at infinity) and the boundary maps use generators of Kleinian groups. Here the endpoints of geodesics can be described by complex continued fractions, which have been studied from the number theoretic perspective by Doug Hensley, S. G. Dani, and Arnaldo Nogueira, among others. In this thesis, a new "partition property" is described, substituting for the cycle property seen in the modular and Fuchsian literature. We
state some results that apply to a wide range of boundary maps satisfying this partition property, and we discuss several specific algorithms in more detail. In many cases, the attractor of the natural extension map can be expressed as a finite union of products in C ⨯ C; this "finite product structure" is explicitly demonstrated for certain algorithms.
Advisors/Committee Members: Svetlana Katok, Dissertation Advisor/Co-Advisor, Svetlana Katok, Committee Chair/Co-Chair, Federico Juan Rodriguez Hertz, Committee Member, Antole Katok, Committee Member, Kyusun Choi, Outside Member.
Subjects/Keywords: geodesic flow; symbolic dynamics; continued fractions; complex numbers; reduction theory; Fuchsian groups
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Record Details
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APA ·
Chicago ·
MLA ·
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CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Zydney, A. J. (2018). Boundary maps and their natural extensions associated with Fuchsian and Kleinian groups. (Thesis). Penn State University. Retrieved from https://submit-etda.libraries.psu.edu/catalog/15102ajz5041
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Zydney, Adam J. “Boundary maps and their natural extensions associated with Fuchsian and Kleinian groups.” 2018. Thesis, Penn State University. Accessed March 03, 2021.
https://submit-etda.libraries.psu.edu/catalog/15102ajz5041.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Zydney, Adam J. “Boundary maps and their natural extensions associated with Fuchsian and Kleinian groups.” 2018. Web. 03 Mar 2021.
Vancouver:
Zydney AJ. Boundary maps and their natural extensions associated with Fuchsian and Kleinian groups. [Internet] [Thesis]. Penn State University; 2018. [cited 2021 Mar 03].
Available from: https://submit-etda.libraries.psu.edu/catalog/15102ajz5041.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Zydney AJ. Boundary maps and their natural extensions associated with Fuchsian and Kleinian groups. [Thesis]. Penn State University; 2018. Available from: https://submit-etda.libraries.psu.edu/catalog/15102ajz5041
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
. 
Open Access Theses and Dissertations