Zhuang Tao: Collet-Eckmann Laminations and Newhouse dynamics

Abstract:  In one-dimensional dynamics, there are quadratic unimodal maps where the orbit of the critical point has a rate of expansion, the so-called "Collet-Eckmann Condition". For maps with the Collet-Eckmann Condition, we cannot find any periodic attractors. In holomorphic dynamics of higher dimensions, the situation is completely different. Following the recent work of Michael Benedicks, Marco Martens, and Liviana Palmisano ( 'Newhouse Laminations', https://arxiv.org/abs/1811.00617 [arxiv.org]), for a dissipative family of holomorphic systems of arbitrary dimensions, we are able to construct a "Collet-Eckmann Lamination" in the parameter space. Each leaf of this lamination is of codimension-1 and each map in the leaf has a critical point with expanding directions and its orbits have chaotic properties. The topological classes of the $\omega-$limit set of the critical point are stable along each leaf of the lamination. We also observe the Newhouse Phenomenon in this lamination. In particular, there are maps in the lamination which has a critical point with the Collet-Eckmann condition and have the coexistence of infinitely many periodic sinks. Moreover, the union of periodic sinks accumulates at the $\omega-$limit set of the critical point. In the parameter space, we can also find leaves in Collet-Eckmann Lamination where a generic point has the Newhouse Phenomenon. The closure of the Newhouse points is the same as the closure of the whole lamination. This is joint work with Marco Martens and Liviana Palmisano.