Moduli spaces and arithmetic dynamics

Includes index

Includes bibliography (p. 127-132) and references

ch. 1 Moduli Spaces Associated to Dynamical Systems 1.1.Dynamical definitions 1.2.Moduli spaces: what they are and why they're useful 1.3.Fine moduli spaces and coarse moduli spaces 1.4.Parameter spaces for dynamical systems 1.5.Moduli spaces for dynamical systems 1.6.Level structure and the uniform boundedness conjecture ch. 2 The Geometry of Dynamical Moduli Spaces 2.1.Introduction to geometric invariant theory (GIT) 2.2.Tools for computing the stable and semistable loci 2.3.Construction of moduli spaces Mnd using GIT 2.4.Multipliers and maps on Md 2.5.M2 is isomorphic to A2 2.6.Uniform bounds for Aut(ø) 2.7.Rationality of M1d 2.8.Special loci in Mnd ch. 3 Dynamical Moduli Spaces-Further Topics 3.1.An application to good reduction over function fields 3.2.Minimal resultants and minimal models 3.3.Dynamics on K3 Surfaces 3.4.An algebraic characterization of Lattes maps Contents note continued: ch. 4 Dynatomic Polynomials and Dynamical Modular Curves 4.1.Dynatomic polynomials 4.2.Dynamical modular curves for z2 + c 4.3.Irreducibility and genus formulas 4.4.Rational points on dynamical modular curves 4.5.Other dynamical modular curves ch. 5 Canonical Heights 5.1.Heights and projective space 5.2.Dynamical canonical heights 5.3.Canonical height zero over function fields 5.4.Local heights and Green functions 5.5.Specialization theorems 5.6.Heights and dominant rational maps 5.7.Canonical heights for regular affine automorphisms 5.8.Canonical heights for K3 dynamics 5.9.Algebraic dynamics and transcendental numbers ch. 6 Postcritically Finite Maps 6.1.Transversality of the PCF locus 6.2.The height of a postcritically finite map 6.3.The invariant measure and the Lyapunov exponent 6.4.Conservative maps 6.5.A dynamical Andre-Oort conjecture ch. 7 Field of Moduli and Field of Definition Contents note continued: 7.1.Twists, automorphisms, and cohomology 7.2.Fields of definition and field of moduli 7.3.Tools for determining when FOM = FOD.

Summary
This monograph studies moduli problems associated to algebraic dynamical systems. It is an expanded version of the notes for a series of lectures delivered at a workshop on Moduli Spaces and the Arithmetic of Dynamical Systems at the Bellairs Research Institute, Barbados, in 2010. The author's goal is to provide an overview, with enough details and pointers to the existing literature, to give the reader an entry into this exciting area of current research.