On the geometry of the parameter space of Hénon like maps and its zoology, from a semi-local linearization technique
speaker: PIerre Berger (CNRS Paris 13)abstract: Hénon like maps appear naturally in the unfolding of a non-degenerate homoclinic tangency, from a renormalization Theorem of Palis-Takens. We will introduce a semi-local linearization technique which provides a linear chart. It enables to perform a revised version of Palis-Takens renormalization.
As applications, it implies that there exist parameters in the Hénon family for which there exist exactly two attractors, which can be a pair of sinks or a pair of one sink with a stochastic attractor ; answering to a question of Lyubich.
It proves also a numerical observation of Milnor stating that the parameter space $(a,b)$ of quadratic maps $g(x)=(x2+a)2+b$ is (a Milnor Swallow) somehow contained in the parameter space of the Hénon family $f(x,y)=(x2+a+y,-bx)$. This implies a negative answer to a question of van Strien.
This result is also useful to prove that the Hausdorff co-dimension of Newhouse phenomena in the unfolding of a homoclinic tangency is at most 12. The latter result is a joint work with Jacopo de Simoi.
timetable:
Mon 5 May, 16:00 - 17:00, Aula Dini
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