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 . The critical points are not required to verify a non-flatness condition, so the results are applicable to 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.
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 . 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.
Keywords: Lyapunov exponent, Pesin theory, absolutely continuous invariant measures, interval dynamics, flat critical points.
Mot clés : exposant de Lyapunov, théorie de Pesin, mesures invariantes et absolument continues, dynamique sur l’intervalle, points critiques plats.
Dobbs, Neil 1
@article{AIF_2014__64_2_571_0, author = {Dobbs, Neil}, title = {On cusps and flat tops}, journal = {Annales de l'Institut Fourier}, pages = {571--605}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {64}, number = {2}, year = {2014}, doi = {10.5802/aif.2858}, mrnumber = {3330915}, zbl = {06387285}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2858/} }
TY - JOUR AU - Dobbs, Neil TI - On cusps and flat tops JO - Annales de l'Institut Fourier PY - 2014 SP - 571 EP - 605 VL - 64 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2858/ DO - 10.5802/aif.2858 LA - en ID - AIF_2014__64_2_571_0 ER -
Dobbs, Neil. On cusps and flat tops. Annales de l'Institut Fourier, Volume 64 (2014) no. 2, pp. 571-605. doi : 10.5802/aif.2858. https://aif.centre-mersenne.org/articles/10.5802/aif.2858/
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