[Mesures invariantes absolument continues hyperboliques pour les applications unidimensionnelles de classe $C^r$]
For $r > 1$, we show, using the Ledrappier–Young entropy characterization of SRB measures for non-invertible maps, that if a $C^r$ map $f$ of the interval or the circle has its Lyapunov exponent greater than $\frac{1}{r} \log \Vert f^{\prime } \Vert _{\infty }$ on a set $A$ of positive Lebesgue measure, then it admits hyperbolic ergodic invariant measures that are absolutely continuous with respect to the Lebesgue measure. We also show that the basins of these measures cover $A$ Lebesgue-almost everywhere.
Pour $r > 1$, nous montrons, en utilisant la caractérisation entropique de Ledrappier-Young des mesures SRB pour les applications non inversibles, que si $f$ est une application $C^r$ de l’intervalle ou du cercle dont l’exposant de Lyapunov est strictement supérieur à $\frac{1}{r} \log \Vert f^{\prime } \Vert _{\infty }$ sur un ensemble $A$ de mesure de Lebesgue positive, alors elle admet au moins une mesure invariante ergodique hyperbolique qui est absolument continue par rapport à la mesure de Lebesgue. Nous montrons également que les bassins de ces mesures recouvrent $A$ Lebesgue-presque partout.
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Keywords: Dynamical systems, SRB measures, Yomdin theory, entropy, Lyapunov exponent
Mots-clés : Systèmes dynamiques, mesures SRB, théorie de Yomdin, entropie, exposant de Lyapunov
Alexandre Delplanque 1
CC-BY 4.0
Alexandre Delplanque. Hyperbolic absolutely continuous invariant measures for $C^r$ one-dimensional maps. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1419-1462. doi: 10.5802/jep.351
@article{JEP_2026__13__1419_0,
author = {Alexandre Delplanque},
title = {Hyperbolic absolutely continuous invariant measures for $C^r$ one-dimensional maps},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1419--1462},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.351},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.351/}
}
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