ICERM Workshop: Moduli Spaces Associated to Dynamical Systems

Moduli Spaces Associated to Dynamical Systems (April 16-20, 2012)

This workshop will bring together dynamicists, number theorists, and algebraic geometers to study the geometry and arithmetic of dynamical moduli spaces. The set Rat_dⁿ of rational degree d self-maps of Pⁿ has a natural structure as an affine variety. The dynamical moduli space M_dⁿ is the quotient of Rat_dⁿ by the conjugation action of the group PGL_n+1. Problems to be investigated include the geometry of M_dⁿ, the distribution of special maps such as post-critically finite maps in M_dⁿ, dynamical modular curves associated to one-parameter families of maps with a marked point of period N, and degeneration of families of maps and the associated points on the boundary of moduli space. A tutorial session will be held the week before this workshop.

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.