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

Abstract

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 !

44 pages

Cited by in corpus (5)

• Finite Resolution Dynamics
• Uniform expansivity outside the critical neighborhood in the quadratic family
• Nonuniformly expanding 1d maps with logarithmic singularities
• Rigorous numerics for critical orbits in the quadratic family
• Rigorous computation of expansion in one-dimensional dynamics