Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps

Details Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps

2005-08-16

arxiv.org/abs/math/0508302v3

Mathematics

Dynamical Systems

44 pages

Scientific paper

10.1088/0951-7715/19/7/013

We formulate and prove a Jakobson-Benedicks-Carleson type theorem on the occurence of nonuniform hyperbolicity (stochastic dynamics) in families of one-dimensional maps, based on "computable starting conditions" and providing "explicit, computable," lower bounds for the measure of the set of selected parameters. As a first application of our results we show that the set of parameters corresponding to maps in the quadratic family f_{a}(x) = x^{2}-a which have an absolutely continuous invariant probability measure is at least 10^-5000 !

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