# ICERM Workshop: Moduli Spaces Associated to Dynamical Systems

The set of rational self-maps of Pn of degree d, which is denoted Ratdn, has a natural structure as an affine variety. The group PGLn+1 acts by conjugation on Ratdn, and the quotient space is the dynamical moduli space Mdn.

Problems

Problem 1: The Geometry of Mdn. It is known that M21 is isomorphic to the affine plane and that Md1 is a rational variety, but many fundamental questions remain. A major goal of the workshop will be to study the geometry of Mdn 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 Md1 in both the complex and the p-adic topologies.

Problem 3: Dynamical Modular Curves. A one-parameter family of maps, for example fc(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.

**Submitted by Anonymous |

**21 / Oct / 2011