Topology and Dynamics on Metric Spaces and Beyond

Chaotic systems exhibit the "butterfly effect": the future path depends sensitively on their starting points, so that a butterfly flapping its wings can set off a tornado on the other side of the world. How can we quantify this sensitivity? This project will develop new notions of how chaotic systems expand and stretch in different directions. Broader impacts of the project are through mentoring of student research, with the project's particular emphasis on concrete, physical models of the resulting fractal geometry, with its surprising and intriguing patterns. In more detail, the project will develop the theory of "topological Lyapunov exponents", a spectrum of rates of expansion for expanding topological dynamical systems. Unlike the older entropy, these new rates control how fast nearby points diverge, measuring distance growth rather than volume growth. Unlike the original Lyapunov exponents, the new rates are defined for topological systems without reference to a smooth structure or notion of differentiation. Nevertheless the topological Lyapunov exponents are interesting and new for smooth systems as well, for instance recovering core entropy for polynomials in the Mandelbrot set. 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.