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

Stefano Luzzatto·Hiroki Takahasi
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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 !

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Stefano Luzzatto, Hiroki Takahasi. 2005-10-07. Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps. https://doi.org/10.1088/0951-7715%2F19%2F7%2F013

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