Conjugacy and entropy of piecewise mobius contact deformations

Abstract

Random matrix products of 2 x 2 matrices may be thought of in a dynamical system setting as iterations functions of mobius maps, with matrix multiplication replacing composition of functions. Sometimes branches of mobius maps may be restricted to form classical dynamical systems. One such family is the tent family about which much is known. Similar properties are investigated for a two parameter family of symmetric piecewise mobius maps (which include the tent family). Using kneading theory, symbolic dynamics, and other techniques, A parameter space is found which foliates into curves of constant kneading sequence on which maps may be pairwise conjugate depending on if the maps restricted to a (forward invariant) core interval are transitive. Investigations into iterated function systems given by the inverse branches of the symmetric family are made by defining a shift on two symbols (depending upon some interval of definition) that models the iterated function system. Continuous deformations of the interval are made and properties of entropy are found. In some cases entropy of the shift space is continuous as the interval is deformed, while in other cases there are discontinuities.