Institute for Mathematical Sciences

Preprint ims01-15

Artur Avila, Mikhail Lyubich and Welington de Melo
Regular or stochastic dynamics in real analytic families of unimodal maps

Abstract: In this paper we prove that in any non-trivial real analytic family of unimodal maps, almost any map is either regular (i.e., it has an attracting cycle) or stochastic (i.e., it has an absolutely continuous invariant measure). To this end we show that the space of analytic maps is foliated by codimension-one analytic submanifolds, ``hybrid classes''. This allows us to transfer the regular or stochastic property of the quadratic family to any non-trivial real analytic family.
View ims01-15 (PDF format)