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Summary: QUASISYMMETRIC ROBUSTNESS OF THE COLLET-ECKMANN CONDITION IN THE
QUADRATIC FAMILY
ARTUR AVILA AND CARLOS GUSTAVO MOREIRA
Abstract. We consider quasisymmetric reparametrizations of the parameter space of the quadratic family. We prove that the set
of quadratic maps which are either regular or Collet-Eckmann with polynomial recurrence of the critical orbit has full Lebesgue
measure.
1. Introduction
Here we consider the quadratic family, fa = a-x2
, where -1/4 a 2 is the parameter. In [AM1], a thorough understanding
of the dynamics of typical (with respect to Lebesgue measure) quadratic maps was obtained. More specifically, it was shown
that a typical quadratic map is either regular (with a periodic attractor) or Collet-Eckmann (positive Lyapunov exponent of the
critical value) with polynomial recurrence of the critical orbit. The first possibility corresponds to a hyperbolic deterministic
setting, with the well known good properties of hyperbolic systems. The second is a particularly well studied case of non-
uniformly hyperbolic chaotic dynamics: in the 90's such maps were shown to possess many hyperbolic-like properties like
stochastic stability, exponential decay of correlations and others ([KN], [Y], [BV] and [BBM]). In particular it was possible to
answer affirmatively Palis Conjecture [Pa] for the quadratic family.
It was shown in [ALM] that the parameter space of general analytic families of unimodal maps (with negative Schwarzian
derivative) can be related to the parameter space of quadratic maps through a quasisymmetric `holonomy map'. It becomes
then feasible to transfer results from the quadratic family to other families, but there is one obstruction: quasisymmetric maps
are not absolutely continuous.
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