Collaborative Research: Floer Theory and Topological Entropy

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

Hamiltonian systems constitute a broad class of dynamical systems where energy dissipation can be neglected. For example, the planetary motion in celestial mechanics, the flow of an incompressible ideal fluid and the motion of a charged particle in an electro-magnetic field are usually treated as Hamiltonian dynamical systems. Topological entropy is an important invariant of a dynamical system, measuring its complexity and originating in physics and information theory. The PIs will develop new methods and tools to study topological entropy of Hamiltonian dynamical systems, utilizing ideas from topological data analysis. Conversely, this research has a potential to contribute to the field of topological data analysis and applied questions including image and pattern recognition. The work involves integration of research, education and training young scientists. It will have impact in the areas of higher education and dissemination of knowledge, within the field and to a wider scientific community, and it will increase participation of individuals from underrepresented groups in mathematics. On a more technical level, the main theme of the project is the interaction between Floer theory and symplectic topology on one side and Hamiltonian dynamics and, in particular, topological entropy on the other. The PIs will study topological entropy of compactly supported Hamiltonian diffeomorphisms and certain Reeb flows from the perspective of Floer theory. The project builds on the PIs’ recent work and focuses on barcode entropy introduced by the PIs, which is a Floer theoretic counterpart of topological entropy and is closely related to it. The key new and distinguishing feature of the PIs’ approach to Floer theoretic aspects of topological entropy is that barcode entropy is based on neither exponential growth of Floer homology – there is no growth in the Hamiltonian setting – nor on topological properties of the map such as the growth of free homotopy classes of periodic orbits. The PIs will also study the behavior of the gamma-norm under iterations in the Hamiltonian or contact setting. Most of the projects will require developing new techniques applicable to other questions, and interactions with areas outside symplectic geometry and dynamics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.