Georgia Institute of Technology College of Sciences

Given a holomorphic map of C^m to itself that fixes a point, what happens to points near that fixed point under iteration? Are there points attracted to (or repelled from) that fixed point and, if so, how? We are interested in understanding how a neighborhood of a fixed point behaves under iteration. In this talk, we will focus on maps tangent to the identity. In dimension one, the Leau-Fatou Flower Theorem provides a beautiful description of the behavior of points in a full neighborhood of a fixed point. This theorem from the early 1900s continues to serve as inspiration for this study in higher dimensions. In dimension 2 our picture of a full neighborhood of a fixed point is still being constructed, but we will discuss some results on what is known, focusing on the existence of a domain of attraction whose points converge to that fixed point.

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.

We provide several explicit examples of 3D quasiperiodic linear skew-products with simple Lyapunov spectrum, that is with 3 different Lyapunov multipliers, for which the corresponding Oseledets bundles are measurable but not continuous, colliding in a measure zero dense set.

Theoretical aspects: If a smooth dynamical system on a compact invariant set is structurally stable, then it has the shadowing property, that is, any pseudo (or approximate) orbit has a true orbit nearby. In fact, the system has the Lipschitz shadowing property, that is, the distance between the pseudo and true orbit is at most a constant multiple of the local error in the pseudo orbit. S. Pilyugin and S. Tikhomirov showed the converse of this statement for discrete dynamical systems, that is, if a discrete dynamical system has the Lipschitz shadowing property, then it is structurally stable. In this talk this result will be reviewed and the analogous result for flows, obtained jointly with S. Pilyugin and S. Tikhomirov, will be described. Numerical aspects: This is joint work with Brian Coomes and Huseyin Kocak. A rigorous numerical method for establishing the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous system of ordinary differential equations is presented. Given a suitable approximate connecting orbit and assuming that a certain associated linear operator is invertible, the existence of a true connecting orbit near the approximate orbit and for a nearby parameter value is proved provided the approximate orbit is sufficiently ``good''. It turns out that inversion of the operator is equivalent to the solution of a boundary value problem for a nonautonomous inhomogeneous linear difference equation. A numerical procedure is given to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse (the latter determines how ``good'' the approximating orbit must be).