TOPOLOGICAL-ENTROPY OF POLYGON EXCHANGE TRANSFORMATIONS AND POLYGONALBILLIARDS

We study the topological entropy of a class of transformations with mi ld singularities: the generalized polygon exchanges. This class contai ns, in particular, polygonal billiards. Our main result is a geometric estimate, from above, on the topological entropy of generalized polyg on exchanges. One of the applications of our estimate is that the topo logical entropy of polygonal billiards is zero. This implies the subex ponential growth of various geometric quantities associated with a pol ygon. Other applications are to the piecewise isometries in two dimens ions, and to billiards in rational polyhedra.