On cusps and flat tops

Dobbs, Neil

Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 571-605.

La théorie de Pesin est développée pour une classe d’applications de l’intervalle, lisses par morceaux. On n’exclut ni des singularités de la dérivée, ni que les points critiques soit plats. On prend comme hypothèse que la dérivée satisfasse à une condition liée à celle de la régularité Hölder.

Nos résultats s’appliquent à des transformations de l’intervalle de classe $C^{1+ϵ}$. Comme conséquence, on démontre l’absence de mesure de probabilité invariante et absolument continue par rapport à la mesure de Lebesgue, lorsque les points critiques sont trop plats. Cela étend un résultat de Benedicks et Misiurewicz.

Non-invertible Pesin theory is developed for a class of piecewise smooth interval maps which may have unbounded derivative, but satisfy a property analogous to $C^{1+ϵ}$. The critical points are not required to verify a non-flatness condition, so the results are applicable to $C^{1+ϵ}$ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.

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