Non-periodic bifurcations of one-dimensional maps

• Autores: Vanderlei Horita, Nivaldo Muniz, Paulo Rogério Sabini
• Localización: Ergodic theory and dynamical systems, ISSN 0143-3857, Vol. 27, Nº 2, 2007, págs. 459-492
• Idioma: inglés
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• Resumen

  • We prove that a ¿positive probability¿ subset of the boundary of ¿{uniformly expanding circle transformations}¿ consists of Kupka¿Smale maps. More precisely, we construct an open class of two-parameter families of circle maps $(f_{a,\theta})_{a,\theta}$ such that, for a positive Lebesgue measure subset of values of $a$, the family $(f_{a,\theta})_\theta$ crosses the boundary of the uniformly expanding domain at a map for which all periodic points are hyperbolic (expanding) and no critical point is pre-periodic. Furthermore, these maps admit an absolutely continuous invariant measure. We also provide information about the geometry of the boundary of the set of hyperbolic maps.