An Afternoon on

Hénon Dynamical Systems

Thursday 12 March, 2015,
Huxley Building, Imperial College London.

14:00-15:00 Shin Kiriki (Tokai University) Huxley 130, Takens' last problem and existence of non-trivial wandering domains

15:05-16:05 Zin Arai (Hokkaido University) Huxley 130, On the monodromy and bifurcations of the Hénon map

16:05-16:25 refreshment break in 5th floor common room

16:30-17:30 Remus Radu (Stony Brook University) Huxley 145, Semi-parabolic Hénon maps and perturbations


Takens' last problem and existence of non-trivial wandering domains (Shin Kiriki, Tokai University)

We give an answer to a version of the open problem of F. Takens 2008 which is related to historic behavior of dynamical systems. To obtain the answer, we show the existence of non-trivial wandering domains near a homoclinic tangency, which is conjectured by Colli and Vargas 2001. Concretely speaking, it is proved that any Newhouse open set in the C^r topology of two-dimensional diffeomorphisms with 2 \leq r < \infty is contained in the closure of the set of diffeomorphisms which have non-trivial wandering domains whose forward orbits have historic behavior. Moreover, this result implies an answer in the C^r category to one of the open problems of van Strien 2010 which is concerned with wandering domains for Hénon family.

On the monodromy and bifurcations of the Hénon map (Zin Arai, Hokkaido University)

We discuss the structure of the parameter space of the Hénon family. Our main tool is the monodromy representation that assigns an automorphism of the full shift to each loop in the hyperbolic parameter locus of the complex Hénon family. We show that the monodromy carries the information of the bifurcations taking place inside the loop, and this enables us to construct pruning fronts, a generalization of kneading theory to the real Hénon family. Furthermore, assuming that there exist infinitely many non-Wieferich prime numbers (it suffices to assume "abc conjecture"), we show that monodromy automorphisms must satisfy a certain algebraic condition, which imposes geometric restrictions on the structure of the parameter space.

Semi-parabolic Hénon maps and perturbations (Remus Radu, Stony Brook University)

We study the global dynamics of complex Hénon maps with a semi-parabolic fixed point that arise as small perturbations of a quadratic polynomial p with a parabolic fixed point. We prove that this family of semi-parabolic Hénon maps is structurally stable on the sets J and J^+; the Julia set J is homeomorphic to a quotiented solenoid (hence connected), while the Julia set J^+ inside a polydisk is a fiber bundle over the Julia set of the polynomial p. We then exhibit certain paths in parameter space for which the semi-parabolic structure can be deformed into a hyperbolic structure, and show that the parametric region of semi-parabolic Hénon maps with small Jacobian lies in the boundary of a hyperbolic component of the Hénon connectedness locus. This is joint work with Raluca Tanase.