Dynamics Seminar

Event Information Hénon maps with semi-indifferent fixed points
15:10 on Monday September 28, 2015
16:00 on Monday September 28, 2015
BA6183, Bahen Center, 40 St. George St.
Remus Radu

Stony Brook University

We consider complex Hénon maps which have a semi-indifferent fixed point with eigenvalues $\lambda$ and $\mu$, where $|\lambda|=1$ and $|\mu|<1$. At a semi-parabolic parameter (i.e. when $\lambda$ is a root of unity) we have a good understanding of this family: for small Jacobian, the dynamics of the Julia set of the Hénon map fibers over the dynamics of a certain polynomial Julia set (joint work with R. Tanase). The situation when $\lambda=\exp(2\pi i \alpha)$ and $\alpha$ is irrational is more complex as it depends on the arithmetic properties of $\alpha$. When $\alpha$ is the golden mean, we show that the H\'enon map with small Jacobian has a Siegel disk whose boundary is homeomorphic to a circle. The proof is based on renormalization of commuting pairs and it is joint work with D. Gaydashev and M. Yampolsky. If $\alpha$ is not Brjuno we prove the existence of a "hedgehog" for holomorphic germs of $(\mathbb{C}^2,0)$ using topological techniques (joint work with T. Firsova, M. Lyubich, and R. Tanase).