Partially Hyperbolic Systems: Closed Periodic Curves and Closed Orbits

The present work is about partially hyperbolic dynamics on closed 3-manifolds and closed periodic curves. Partially hyperbolic systems are dynamical systems characterized by a dominated splitting of their tangent bundle into directions of contraction, expansion and intermediate behavior. The prime candidates of study are Anosov flows and partially hyperbolic diffeomorphisms. In recent years, these systems have gained a lot of interest, mainly in dimension 3, where there is lot of machinery available. In the realm of Anosov flows, the orbit structure of these systems plays a major role in understanding their properties and even helping in their classification. For skewed R-covered Anosov flows whose weak stable foliation is transversely orientable, it was shown by Fenley [7] that any two distinct freely homotopic periodic orbits are always isotopic. We were interested in knowing whether this result was also true for non R-covered Anosov flows. Our first main result is to extend this result to a class of systems generalizing Anosov flows, called topological Anosov flows. This is theorem 34. In contrast with Anosov flows, partially hyperbolic diffeomorphisms may not exhibit a very desirable property called dynamical coherence. This property means that the center direction of the system integrates into a foliation. The first example of a non dynamically coherent partially hyperbolic diffeomorphism in dimension 3 was constructed by Hertz, Hertz, and Ures [37]. The existence of non differentiable points in curves along the center direction, is what makes the system not dynamically coherent. Quite commonly, important invariant sets of regularity no more than C 0 appear in the study of dynamical systems whose regularity is C 1 or better. Seeing as periodic orbits play an important role, we wonder whether a continuous embedded (possibly non differentiable) closed periodic curve, under a partially hyperbolic diffeomorphism could be "tangent" (see remark 11.2.1) to the center direction. We show this is indeed the case in our second main result, theorem 53.

Grantor dc:publisher

Florida State University

Year dc:date

2024

Contributors dc:contributor

• Molina Gonzalez, Braulio (author)
• Fenley, Sérgio Roberto (professor directing dissertation)
• Reina, Laura (university representative)
• Heil, Wolfgang H. (committee member)
• Bowers, Philip L., 1956- (committee member)
• Florida State University (degree granting institution)
• College of Arts and Sciences (degree granting college)
• Department of Mathematics (degree granting department)

• Mathematics

Language dc:language

English

Identifier

fsu:927999
iid: MolinaGonzalez_fsu_0071E_18677

OAI identifier oai:identifier

oai:diginole.lib.fsu.edu:fsu_927999