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Fractal billiards, flat surfaces, IETs

C. Johnson and I are currently investigating the flow on a fractal flat surface. Related to this is another joint project concerning the existence of a saddle connection on a fractal billiard table that connects two elusive points of the fractal billiard table. The definition of a fractal flat surface is still not concretely stated, but in specific instances can be formally defined.  In addition to such a surface, we have defined a fractal interval exchange transformation for the T-fractal flat surface.

My joint work with Joe P. Chen on a project that extends the work of Jeremy Tyson and Estibalitz Durand-Cartagena [Du-CaTy] has recently appeared in the Journal of Mathematical Analysis & Applications. The project involves describing periodic orbits of a self-similar Sierpinski carpet billiard table.

I have worked with M. L. Lapidus on various papers on the topic of fractal billiards. Specifically, an article with M. L. Lapidus and R. L. Miller consists of determining periodic orbits of the T-fractal billiard table. An eventual goal is understanding whether or not there are equidistributed orbits on the T-fractal billiard table. An initial step in this direction is to understand whether or not every orbit with an irrational direction is recurrent in the T-fractal billiard.