Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, RI
This workshop will bring together researchers working in classical complex dynamics and in the newer area of p-adic (nonarchimedean) dynamics. It will promote interactions between the two groups by highlighting the similarities and differences between complex and p-adic dynamics. In particular, it will address Berkovich space, whose introduction has greatly enhanced the exchange of ideas between complex and p-adic dynamics.
Problem 1: Potential Theory and Invariant Measures. In the past decade, a number of researchers have developed a robust potential theory on Berkovich spaces and used it to construct an invariant measure associated to each rational map acting on Berkovich projective line, which is in some respects analog to the maximal entropy measure in complex dynamics. It is known to be ergodic and supported on the Julia set, but many fundamental questions remain.
Problem 2: Equidistribution Theorems. Various special sets of points (periodic, preperiodic, small height, Galois conjugates, etc.) have been shown to be equidistributed with respect to the maximal entropy measure in the Riemann sphere and/or to the analog measure on Berkovich projective line. These equidistribution results have had a number of important applications. A focus of the workshop will be to formulate and prove additional equidistribution results and give further applications.
Problem 3: Dynamics of Fatou Components. In 2000, Rivera-Letelier classified the possible dynamics on periodic components of the Berkovich-space Fatou set of a rational function, a classification that is analogous to the complex results of Fatou and Julia. Many questions remain open, including bounds for the number of periodic cycles of quasiperiodicity domains and the existence of wandering Fatou domains over the p-adic rationals.
Problem 4: Transversality in Complex Dynamics. Thurston's rigidity theorem gives a topological characterization of critically finite rational maps. Recent work of Epstein on an infinitesimal version of Thurston's theorem has an underlying arithmetic flavor. The workshop will provide an opportunity to explore these powerful new analytic/algebraic tools for studying both complex and p-adic dynamics.
Problem 5: The Local Connectivity Conjecture. This longstanding and fundamental conjecture asserts that the Mandelbrot set is locally connected. Its resolution would have major implications, including the density of hyperbolic maps in moduli space. Might there be tools from arithmetic dynamics which could be useful for studying this fundamental conjecture?