Volume 389, 2005
Phase-Parameter Relation and Sharp Statistical Properties
for General Families of Unimodal Maps
Artur Avila* and Carlos Gustavo Moreira**
ABSTRACT. We obtain precise estimates relating the phase space and the pa-
rameter space of analytic families of unimodal maps, which generalize the case
of the quadratic family obtained in
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 con-
clude 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 ap-
plies also without the quasiquadratic assumption. Those ideas lead to a proof
of a theorem of Shishikura: the set of non-renormalizable parameters in the
boundary of the Mandelbrot set has zero Lebesgue measure. Further applica-
tions of those results can be found in
A unimodal map is a smooth (at least C
I-+ I, where I
JR. is an
interval, which has one unique critical point c E int I which is a maximum. Let
us say that
is regular if it has a quadratic critical point, is hyperbolic and its
critical point is not periodic or preperiodic. By a result of Kozlovski
of regular maps coincides with the set of structurally stable unimodal maps, and
it follows that the set of regular maps is open and dense in all smooth (and even
A central problem in dynamical systems is to give a good statistical description
of "typical systems", for some reasonable measure-theoretical notion of typical:
Ideally (according to the Palis Conjecture [P]) the set of systems with a good
statistical description should correspond to a Lebesgue full measure set in (a large
set of) parametrized families.
In order to establish such a picture, one should understand well how the dynam-
ics varies with the parameter. In one-dimensional dynamics, a basic approach has
been to investigate thoroughly the dynamics of an individual map and show that
some of its properties are reflected on the nearby parameter space. As formulated
2000 Mathematics Subject Classification. Primary 37E05, Secondary 37F45.
*Partially supported by the French-Brazilian Agreement in Mathematics.
••Partially supported by Faperj and CNPq, Brazil.
@ 2005 American Mathematical Society