The Art of Iterating Rational Functions over Finite Fields

The field of complex dynamical systems generated by iteration of polynomials and rational functions is a classical area of mathematics with a rich history and a wide variety of results. Recently, there has been substantial interest in arithmetical dynamical systems (ADS), meaning the iteration of rational functions over fields of number-theoretic interest. Although isolated results regarding ADS have been proven throughout the twentieth century, it was only in the 1990’s that ADS were identified as a field of study. The past two decades have seen an explosion of work in this topic, in which fundamental problems have been solved, new questions and conjectures have been posed, and connections have been forged with a great many different areas of pure and applied mathematics. Indications of the fast growing significance of ADS also include the 2010 Mathematics Subject Classification, which contains a new section, 37Pxx, devoted to arithmetic and non-archimedean dynamical systems, whose subsections include 37P05 (Polynomial and rational maps), 37P35 (Arithmetic properties of periodic points), and 37P55 (Arithmetic dynamics on general algebraic varieties), all of which are directly related to the proposed workshop. The significance of and interest in this topic has also been evidenced over the last several years by the large number of international meetings and workshops in this area, which have attracted experts from dynamical systems as well as experts from algebra and number theory. From this list we note in particular the program on Dynamical Systems that was held at ICERM in Spring 2012, which was focused on complex dynamics, p-adic dynamics, global arithmetic dynamics, and moduli spaces associated to dynamical systems. However, despite this recent profusion of scientific activity in related areas, little progress has been made on Dynamical Systems over Finite Fields (DFF). The present exciting and challenging mathematical problems in DFF are of an intricate algebraic and number theoretic flavour whose study requires deep mathematical and computational tools. This area of research focuses on the construction of non-classical dynamical systems over finite fields and the study of atypical behaviour which is not present in standard constructions from complex dynamical systems. Such novel constructions and the classification of their anomalous behaviour have led to innovative solutions to problems in cryptography, biological and physical systems. Many important questions regarding the orbits of DFF remain wide open, including questions on the number of aperiodic points, the size of orbit intersections (in linearly disjoint DFF), cycle lengths, and so on. Furthermore, the problems proposed as the goal of this proposal have barely been touched upon by other meetings. On the other hand,