Dynamics and Translation Surfaces

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Low complexity dynamical systems include symbolic systems such as Sturmian sequences and the Morse sequence; horocycle flows and other unipotent flows on spaces of algebraic origin; rational polygonal billiards and related systems such as linear flows on translation surfaces and interval exchange transformations. In this proposal we consider questions which deal with the relations between these different areas of low complexity dynamics. If we consider a billiard trajectory and keep track of the sides that it hits we can interpret the result as a symbolic system. We investigate symbolic systems that arise this way. The study of such systems gives us a common ground in which symbolic techniques and polygonal billiard techniques can both be applied. The collection of translation surfaces of a fixed topological type forms a topological space which we think of as a moduli space. This space has a natural unipotent flow on it which is usually called the horocycle flow. We consider to what extent does this flow behave like the unipotent flows described by Ratner's classification theorem. In particular we consider the extent to which techniques that work in the homogeneous case do and do not work in the moduli space case.In physics, biology, economics and many other fields one studies the long term behavior of mathematical models of systems that change with time. If these systems are deterministic and autonomous, that is to say isolated from their environment, then they fall in the domain of the field of dynamical systems. A fundamental discovery of this field is that there are common elements of behavior for systems that arise in different areas. There are for example common features to "chaotic" systems. In this proposal we study a class of systems which we call low complexity by contrast with typical dynamical systems which are "chaotic" or "high complexity". This proposal deals with three families of low complexity dynamical systems and the relations between them.