The Symmetry of Chaos

PLEASE NOTE: This is a WEBINAR

We will describe the properties of dynamical systems that:

(1) possess symmetry

(2) exhibit chaotic behavior

In an initial study of such systems, Miranda and Stone projected the Lorenz attractor in a 2 to 1 locally diffeomorphic way to the ``proto''-Lorenz attractor. Then they ``lifted'' this attractor back up to n-fold covers in a locally diffeomorphic way using properties of the rotation group Cn and some complex analysis. We describe the interaction of symmetry groups with equivariant (symmetric) dynamical systems and show how invariant polynomials and an integrity basis are used to construct image dynamical systems.  There is an unexpected richness in ``lifting'' invariant dynamical systems up to equivariant dynamical systems, as different groups anddifferent singular sets can be used to construct locally diffeomorphic but topologically inequivalent covering dynamical systems. Different covers are labeled by distinct values of topological indices. These ideas will be illustrated with lots of pictures.