Dynamics and Universality of Unimodal Mappings with Infinite Criticality

Details Dynamics and Universality of Unimodal Mappings with Infinite Criticality Dynamics and Universality of Unimodal Mappings with Infinite Criticality

2003-06-02

arxiv.org/abs/math/0306033v2

Mathematics

Dynamical Systems

misprints and minor errors are corrected, accepted by the Communications in Mathematical Physics

Scientific paper

We consider infinitely renormalizable unimodal mappings with topological type which is periodic under renormalization. We study the limiting behavior of fixed points of the renormalization operator as the order of the critical point increases to infinity. It is shown that a limiting dynamics exists, with a critical point that is flat, but still having a well-behaved analytic continuation to a neighborhood of the real interval pinched at the critical point. We study the dynamics of limiting maps and prove their rigidity. In particular, the sequence of fixed points of renormalization for finite criticalities converges, uniformly on the real domain, to a mapping of the limiting type.

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