One-resonance and doubly connected basins of attraction in dynamics of iterated holomorphic mapsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

A holomorphic map is locally given by its power series expansion. Thus, the local dynamics of the an iterated holomorphic map near a fixed point can often be determined from a finite number of terms of the expansion at the fixed point. Local invariant sets with locally stable dynamics can then be extended via backward images to global objects with the same long-term dynamics. For polynomials on C, stable dynamics near fixed points are well understood and all stable dynamics arise from fixed points. In C2 both local and global stable dynamics still pose many open questions. New types of dynamics at neutral fixed points in C2 arise from non-trivial multiplicative relations of eigenvalues in the linear part, called resonances. In this talk I will present a construction of examples in C2 with a product of eigenvalues equal to 1. In this so-called one-resonant case, we obtain a non-linear projection to one variable, allowing us to construct a doubly connected attracting open set. To show the “hole” cannot be filled to obtain a larger simply connected attracting set, we impose a small divisor condition on the eigenvalues and show that our attracting set is the whole basin of attraction of the fixed point.