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Non-uniformly expanding dynamics in maps with singularities and criticalities. (English) Zbl 0978.37029
A certain dynamical aspect of a subtle class of one-parameter families of interval maps is studied in this work. The maps are odd functions with a discontinuity at the origin, two critical points, $$C^1$$ outside the origin and $$C^2$$ outside the origin and the critical points. They enjoy a concavity property outside the origin. Also, there are conditions on expandicity and periodicity inside a family. The rather technical result implies the existence of a parameter subset of positive Lebesgues measure for whose members the maps have positive Lyapunov exponent and which contains a dynamically important density point. – The motivation for studying these families of maps is their relevance in the context of the 3-dimensional Lorenz system.

##### MSC:
 37E05 Dynamical systems involving maps of the interval 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
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