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?

Problem 1: The Geometry of M_dⁿ. It is known that M₂¹ is isomorphic to the affine plane and that M_d¹ is a rational variety, but many fundamental questions remain. A major goal of the workshop will be to study the geometry of M_dⁿ and the associated moduli spaces in which one adds level structure, for example by adding a marked point of period N or a marked finite orbit of order N. A motivating question is whether the resulting varieties are of general type if N is sufficiently large.

Problem 2: Distribution of Special Points. An example of the type of problem to be considered is the distribution of post-critically finite maps in the moduli space M_d¹ in both the complex and the p-adic topologies.

Problem 3: Dynamical Modular Curves. A one-parameter family of maps, for example f_c(z)=z²+c, with marked points or orbits of order N, yields dynamical modular curves X₀(N) and X₁(N) that are analogous to classical modular curves. A good deal is known about the geometry of these curves, but little has been proven about their arithmetic except for some small values of N. The arithmetic properties of X₀(N) and X₁(N) are closely related to the uniform boundedness conjecture for the families that they parameterize.

Problem 4: The Boundary of Moduli Space. The boundary of a moduli space and a natural method for completing the space are of fundamental importance in understanding the underlying objects and their degenerations. Recent work of Kiwi has used Berkovich space dynamics over Laurent series fields to analyze degenerations of complex dynamical systems. A goal of the workshop is to exploit these non-archimedean methods to answer classical questions about the boundary of dynamical moduli spaces over the complex numbers.