Invertible
maps of the plane form an important class of dynamical systems, both in their
own right and as simple models of more complicated systems. Such maps often exhibit geometric
structures, called homoclinic tangles, that help to organize the transport behavior
of their dynamics. The topology of
homoclinic tangles thus provides a route for understanding the qualitative
transport properties of maps.
Since the original introduction of homoclinic tangles by Poincare over a
hundred years ago, they have received considerable attention. Nevertheless, there seems to be a lack
of general techniques applicable to the wide variety of tangles realized by
physically relevant maps. To
address this deficiency, we introduce a new topological technique, called
homotopic lobe dynamics. This technique
allows one to describe new dynamical features that appear as the map is
iterated for longer and longer times.
Kevin Mitchell
UC-Merced