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Phase-parameter relation and sharp statistical properties for general families of unimodal maps. (English) Zbl 1145.37022

Eells, James (ed.) et al., Geometry and dynamics. International conference in honor of the 60th anniversary of Alberto Verjovsky, Cuernavaca, Mexico, January 6–11, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3851-2/pbk). Contemporary Mathematics 389. Aportaciones Matemáticas, 1-42 (2005).
A unimodal map is a \(C^2\) map of a closed interval that attains its maximum value at a single point in the interior of the interval, and this is its only critical point. The paper studies finite-parameter families of such maps.
The first main result is that a one-parameter analytic family \(f_\lambda\) of unimodal maps satisfying a nontriviality condition (which holds for a dense set of such families) satisfies what is called the “phase-parameter” relation at almost every nonregular parameter \(\lambda_0\). These are precise measure-theoretic estimates relating the phase space of \(f_{\lambda_0}\) near its critical point to the parameter space near \(\lambda_0\). The parameter being regular means that the critical point is quadratic, neither periodic nor preperiodic, but is asymptotic to a periodic orbit, and all periodic orbits are hyperbolic. The second main theorem makes use of this relation to show that for a finite-parameter family of analytic unimodal maps satisfying the same nontriviality condition, for almost every parameter the map is either regular or it has a renormalization that is quasiquadratic (any \(C^3\) map sufficiently close in the \(C^3\) topology is topologically conjugate to a quadratic map).
This leads to several important corollaries. First, it follows from the second theorem that, with the same hypotheses, at almost every nonregular parameter the critical point of the map is polynomially recurrent and there is a hyperbolic growth condition on the derivatives of the iterates of the map on the image of the critical point (Collet-Eckmann condition). The first theorem can then be used to show that the critical point is polynomially recurrent with exponent 1. Second, for any \(k\geq 2\) or \(k=\infty\), a generic finite-parameter family of \(C^k\) unimodal maps is, at almost every parameter, either regular or else is Collet-Eckmann, with its critical point subexponentially recurrent, and has a renormalization conjugate to a quadratic.
An appendix shows how to extend the proof of the second theorem to complex families and to obtain a proof of Shishikura’s theorem that in the family \(p_c(z) = z^2 + c\), the set of \(c\) for which \(p_c\) is neither hyperbolic nor infinitely renormalizable has Lebesgue measure 0.
For the entire collection see [Zbl 1078.37003].

MSC:

37E05 Dynamical systems involving maps of the interval
37E20 Universality and renormalization of dynamical systems
37F25 Renormalization of holomorphic dynamical systems
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
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