Algebraic Geometry of Discrete Dynamics. The case of one variable

Abstract

We argue that discrete dynamics has natural links to the theory of analytic functions. Most important, bifurcations and chaotic dynamical properties are related to intersections of algebraic varieties. This paves the way to identification of boundaries of Mandelbrot sets with discriminant varieties in moduli spaces, which are the central objects in the worlds of chaos and order (integrability) respectively. To understand and exploit this relation one needs first to develop the theory of discrete dynamics as a solid branch of algebraic geometry, which so far did not pay enough attention to iterated maps. The basic object to study in this context is Julia sheaf over the universal Mandelbrot set. The base has a charateristic combinatorial structure, which can be revealed by resultant analysis and represented by a basic graph. Sections (Julia sets) are contractions of a unit disc, related to the action of Abelian $\bb{Z}$ group on the unit circle. Their singularities (bifurcations) are located at the points of the universal discriminant variety.