The main objects of the proposed research are smooth dynamical systems such as diffeomorphisms and flows, as well as smooth actions of higher rank abelian groups. The main goal of the project is to study various rigidity properties for hyperbolic, partially hyperbolic, and non-uniformly hyperbolic systems and actions. Higher rank actions exhibit remarkable properties including rigidity of invariant measures and smooth rigidity. In rank one case, rigidity occurs for some non-typical systems, such as those with high smoothness of invariant foliations or strong pinching. An important role in both areas is played by cocycles. Investigating cohomology of non-commutative cocycles over hyperbolic systems is a major part of the proposed research.
The field of Dynamical Systems studies evolution of mechanical, physical, and mathematical systems over time. It provides applications to other areas of mathematics as well as to many natural sciences such as physics, mechanics, computer science, and biology. This field originated from differential equations and celestial mechanics. It uses mathematical tools from analysis and topology to study qualitative behavior of systems "in the long run". A major part of the project is the study of more complex systems consisting of several individual ones that commute with each other. Such complex systems appear naturally in algebra, geometry, and physics. The PI works for the main regional university which is primarily undergraduate and educates most of the local school teachers.
The main objects of the proposed research were smooth dynamical systems such as diffeomorphisms and flows, as well as smooth actions of higher rank abelian groups.The main goals of the project were to investigate rigidity phenomena in dynamics and to investigate cocycles over dynamical systems and over group actions. The field of Dynamical Systems studies evolution of mechanical, physical, and mathematical systems over time. It provides applications to other areas of mathematics as well as to many natural sciences such as physics, mechanics, computer science, and biology. This field originated from differential equations and celestial mechanics. It uses mathematical tools from analysis and topology to study qualitative behavior of systems "in the long run". A major part of the project was the study of complex systems consisting of several related individual ones. Such ystems appear naturally in algebra, geometry, and physics. The PI obtained smooth classification for genuinely higher rank hyperbolic actions of Z^k (the integer lattice in the k-dimensional Euclidean space) on tori, and more generally on nilmanifolds, under assumptions close to optimal. This means that such actions differ only by a differentiable change of coordinates from the standard models given by algebraic formulas. New methods were introduced to successfully overcome major obstacles. Several results of independent interest were also obtained including a new elliptic type regularity result for functions and exponential decay of correlations for Holder continuous functions. The techniques developed in this research proved very useful and have already been applied to obtain several important results. Cocycles appear naturally in various areas of dynamical systems and have many applications. In smooth dynamics and rigidity an important role is played by linear cocycles that arise from the derivative of the system. The PI obtained significant advances in the study of cohomology of Holder continuous linear cocycles over hyperbolic systems and over accessible partially hyperbolic systems. The PI obtained regularity results for measurable invariant objects for such cocycles and established a structure theorem for cocycles with one Lyapunov exponent. The findings of the PI's research supported by this grant have appeared (and will continue to appear as they are accepted and published) in high quality pear-reviewed journals. These findings have also been presented in various talks at colloquia,conferences, and seminars. The PI ensures dissemination of his research by posting his papers on the ArXiv and by maintaining the PI's personal web page where all his recent papers are freely available. Until August 2012, the PI worked at the University of South Alabama, which is a primarily undergraduate institution. It is the only major university serving the Mobile urban area and the rural region of Alabama and Mississippi Gulf coast, with many students coming from underrepresented groups. The majority of local teachers graduate from the university, so the PI's teaching and research directly affected the community. The PI taught graduate courses, including ones designed for future and current high school teachers pursuing master's degree. The PI served as an Academic Advisor for Mathematics and Mathematics/Secondary Education majors. In the Fall of 2012, the PI joined the Center for Dynamics and Geometry at Penn State, which consists of faculty, post-docs, and graduate students. The PI immediately became involved in various activities of the center. He was an organizer of the Penn State Workshop in Dynamical Systems and Related Topics in October 2012 and 2013. The PI actively participated in MASS Program and the Pre-MASS Program at Penn State. The Mathematics Advanced Study Semesters (MASS) is an intensive program for talented and motivated undergraduate students, who are recruited from U.S. colleges and universities and brought to Penn State for the fall semester. For majority of its participants, the program has a formative effect and leads to graduate studies and careers in mathematics or related sciences. Pre-MASS is a similar preparatory program in the spring semester.
