On the growth of local intersection multiplicities in holomorphic dynamics

CDSNS Colloquium
Monday, October 6, 2014 - 11:00am
1 hour (actually 50 minutes)
Skiles 005
School of Mathematics Georgia Inst. Technology
In this talk, we will discuss a question posed by Vladimir Arnold some twenty years ago, in a subject he called "dynamics of intersections." In the simplest setting, the question is the following: given a (discrete time) holomorphic dynamical system on a complex manifold X and two holomorphic curves C and D in X which pass through a fixed point P of the system, how quickly can the local intersection multiplicies at P of C with the iterates of D grow in time? Questions like this arise naturally, for instance, when trying to count the periodic points of a dynamical system. Arnold conjectured that this sequence of intersection multiplicities can grow at most exponentially fast, and in fact we can show this conjecture is true if the curves are chosen to be suitably generic. However, as we will see, for some (even very simple) dynamical systems one can choose curves so that the intersection multiplicities grow as fast as desired. We will see how to construct such counterexamples to Arnold's conjecture, using geometric ideas going back to work of Yoshikazu Yamagishi.