New Directions in Partially Hyperbolic Dynamics

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This project seeks to gain a deeper understanding of chaotic dynamical systems. Chaotic dynamics is abundant in the real world and appears throughout sciences and engineering. It can appear in simple mechanical mechanisms such as a double pendulum or in complicated natural phenomena such as convection currents in the atmosphere. The study of such naturally arising examples is extremely challenging and we are very far from a good understanding of them. The partially hyperbolic dynamical systems that will be studied in this project are simplified mathematical models of chaotic dynamical systems that arise in nature. The emphasis of the project will be on a wide range of problems which are new in the field or have received only marginal attention in the past. Potentially, this research may have impacts in physics, biology, chemistry, engineering and other areas where chaotic dynamical systems arise.Anosov and partially hyperbolic diffeomorphisms are the prime examples of chaotic dynamical systems. In the past decades the study of ergodic and chaotic properties in partially hyperbolic dynamics has flourished, while geometric aspects haven't received as much attention. This will be a wide-ranging program to develop geometric structure theory for partially hyperbolic dynamics. The program includes: (1) further development of the Anosov bundle theory; (2) development of topological and global structural stability for partially hyperbolic diffeomorphisms; (3) initiation of the smooth conjugacy program for 3-dimensional partially hyperbolic diffeomorphisms. Further, the project will to bring partially hyperbolic vision beyond the dominated splitting paradigm and take first steps into the realm of homological and coarse hyperbolicity. Finally, the project will expand work on new examples in partially hyperbolic dynamics. New examples have intrinsic value and can be used as additional ground for the structure theory to be applied and tested. The project will use a diverse blend of dynamics and three and higher dimensional topology.