Dynamics of quadratic polynomials, III Parapuzzle and SBR measures

This is a continuation of notes on the dynamics of quadratic polynomials. In this part we transfer our previous geometric result [L3] to the parameter plane. To any parameter value c (outside the main cardioid and the little Mandelbrot sets attached to it) we associate a "principal nest of parapuzzle pieces". We then prove that the moduli of the annuli between two consecutive pieces grow at least linearly. This implies, using Martens & Nowicki (cf. this volume) geometric criterion for existence of an absolutely continuous invariant measure together with [L2], that Lebesgue almost every real quadratic polynomial is either hyperbolic, or has a finite absolutely continuous invariant measure, or is infinitely renormalizable. In the further papers [L5,L7] we show that the latter set has zero Lebesgue measure, which completes the measure-theoretic picture of the dynamics in the real quadratic family.