Phase-parameter Relation and Sharp Statistical Properties for General Families of Unimodal Maps

We obtain precise estimates relating the phase space and the parameter space of analytic families of unimodal maps, which generalize the case of the quadratic family obtained in [AM1]. This result implies a statistical description of the dynamics of typical analytic quasiquadratic maps which is much sharper than what was previously known: as an example, we can conclude that the recurrence of the critical point is polynomial with exponent one. To complete the picture, we show that typical analytic non-regular unimodal maps admit a quasiquadratic renormalization, so that the previous result applies also without the quasiquadratic assumption. 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 zero Lebesgue measure. Further applications of those results can be found in the companion paper [AM3].