Phase-parameter relation and sharp statistical properties for general families of unimodal maps

Artur Avila; Carlos Gustavo Moreira

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arXiv · math/0306156

Phase-parameter relation and sharp statistical properties for general families of unimodal maps

Artur Avila·Carlos Gustavo Moreira

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Abstract

We obtain estimates relating the phase space and the parameter space of analytic families of unimodal maps. Using those estimates, we show that typical analytic unimodal maps admit a quasiquadratic renormalization. This reduces the study of the statistical properties of typical unimodal maps to the quasiquadratic case which had been studied in \cite {AM2}. The estimates proved here correspond exactly to the Phase-Parameter relation proved in \cite {AM} in the quadratic case, and allows one to obtain sharp estimates on the dynamics of typical unimodal maps which were available only in the quadratic case: as an example we conclude that the exponent of the polynomial recurrence of the critical orbit is exactly one. We also show that those ideas lead to a new proof of a Theorem of Shishikura: the set of non-renormalizable parameters in the boundary of the Mandelbrot set has Lebesgue measure zero. Further applications of those results can be found in the companion paper \cite {AM3}.

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Artur Avila, Carlos Gustavo Moreira. 2003-06-10. Phase-parameter relation and sharp statistical properties for general families of unimodal maps. https://arxiv.org/abs/math/0306156

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