Arek Goetz: Global and local behavior of orbits in the two halfplane map. Computer aided renormalization techniques

Abstract:

We start by defining the most basic piecewise rotation on the plane T. The map T acts as rotation R1 on the upper halfplane and T is rotation R2 on the lower halfplane. We illustrate that the global dynamics of T depends on whether the images of the halfplanes overlap or form a gap. In the third remaining case, that is in the case if T preserves area, the dynamics is much more subtle, and it is not completely understood. Computer graphics suggest a KAM like behaviors for irrational choices of parameters. For rational choices, macroscopic images suggest some mysterious Farry tree and continued fractions. On the microscopic level, when zooming highly nontrivial renormation structures may be observed. We conclude the talk with a discussion on implementing a computer investigation in ways to avoid losing accuracy under iterative zooming.