On the Numerical Construction of Hyperbolic Structures for Complex Dynamical Systems
by Suzanne Lynch Hruska
August 2002
Our main interest is using a computer to rigorously study ε-pseudo orbits for polynomial diffeomorphisms of C2. Periodic ε-pseudo orbits form the ε-chain recurrent set, Rε. The intersection ∩ε > 0 Rε is the chain recurrent set, R. This set is of fundamental importance in dynamical systems.
Due to the theoretical and practical difficulties involved in the study of C2, computers will presumably play a role in such efforts. Our aim is to use computers not only for inspiration, but to perform rigorous mathematical proofs.
In this dissertation, we develop a computer program, called Hypatia, which locates Rε, sorts points into components according to their ε-dynamics, and investigates the property of hyperbolicity on Rε. The output is either “yes”, in which case the computation proves hyperbolicity, or “not for this ε“, in which case information is provided on numerical or dynamical obstructions.
A diffeomorphism f is hyperbolic on a set X if for each x there is a splitting of the tangent bundle of x into an unstable and a stable direction, with the unstable (stable) direction expanded by f (f–1). A diffeomorphism is hyperbolic if it is hyperbolic on its chain recurrent set.
Hyperbolicity is an interesting property for several reasons. Hyperbolic diffeomorphisms exhibit shadowing on R, i.e., ε-pseudo orbits are delta-close to true orbits. Thus they can be understood using combinatorial models. Shadowing also implies structural stablity, i.e., in a neighborhood in parameter space the behavior is constant. These properties make hyperbolic diffeomorphisms amenable to computer investigation via ε-pseudo orbits.
We first discuss Hypatia for polynomial maps of C. We then extend to polynomial diffeomorphisms of C2. In particular, we examine the class of Hénon diffeomorphisms, given by
Ha,c : (x, y) → ( x2 + c – ay, x).
This is a large class of diffeomorphisms which provide a good starting point for understanding polynomial diffeomorphisms of C2. However, basic questions about the complex Hénon family remain unanswered.
In this work, we describe some Hénon diffeomorphisms for which Hypatia verifies hyperbolicity, and the obstructions found in testing hyperbolicity of other examples.