Dimension of weakly expanding points for quadratic maps

Senti, Samuel

doi:10.24033/bsmf.2448

Dimension of weakly expanding points for quadratic maps
[Dimension des points faiblement dilatants pour l'application quadratique]

Senti, Samuel

Bulletin de la Société Mathématique de France, Tome 131 (2003) no. 3, pp. 399-420.

Résumé
Abstract

Pour l’application quadratique réelle $P_{a} (x) = x^{2} + a$ et un $ϵ > 0$ donné, un point $x$ a de bonnes propriétés de dilatation si tout intervale contenant $x$ contient également un voisinage $J$ de $x$ avec $P_{a}^{n} |_{J}$ univalent, avec distortion bornée et $B (0, ϵ) \subseteq P_{a}^{n} (J)$ pour un $n \in ℕ$ . L’ensemble $ϵ$ -faiblement dilatant est l’ensemble des points qui n’ont pas de bonnes propriétes de dilatation. Notons $α$ le point fixe négatif et $M$ le temps de premier retour de l’orbite critique dans $[α, - α]$ . Nous prouvons l’existence d’un ensemble $ℛ$ de paramètres de mesure de Lebesgue positive pour lesquels la dimension de Hausdorff de l’ensemble $ϵ$ -faiblement dilatant est bornée supérieurement et inférieurement par ${log}_{2} M / M + 𝒪 ({log}_{2} {log}_{2} M / M)$ si $ϵ$ est proche de $| α |$ . Pour $ϵ \leq | α |$ quelconque la dimension est de l’ordre de $𝒪 ({log}_{2} | {log}_{2} ϵ | / | {log}_{2} ϵ |) .$ Les constantes ne dependent que de $M$ . Le théorème du Folklore implique alors l’existence d’une mesure de probabilité absolument continue et invariante par $P_{a}$ pour $a \in ℛ$ (théorème de Jakobson).

For the real quadratic map $P_{a} (x) = x^{2} + a$ and a given $ϵ > 0$ a point $x$ has good expansion properties if any interval containing $x$ also contains a neighborhood $J$ of $x$ with $P_{a}^{n} |_{J}$ univalent, with bounded distortion and $B (0, ϵ) \subseteq P_{a}^{n} (J)$ for some $n \in ℕ$ . The $ϵ$ -weakly expanding set is the set of points which do not have good expansion properties. Let $α$ denote the negative fixed point and $M$ the first return time of the critical orbit to $[α, - α]$ . We show there is a set $ℛ$ of parameters with positive Lebesgue measure for which the Hausdorff dimension of the $ϵ$ -weakly expanding set is bounded above and below by ${log}_{2} M / M + 𝒪 ({log}_{2} {log}_{2} M / M)$ for $ϵ$ close to $| α |$ . For arbitrary $ϵ \leq | α |$ the dimension is of the order of $𝒪 ({log}_{2} | {log}_{2} ϵ | / | {log}_{2} ϵ |) .$ Constants depend only on $M$ . The Folklore Theorem then implies the existence of an absolutely continuous invariant probability measure for $P_{a}$ with $a \in ℛ$ (Jakobson’s Theorem).

MR Zbl | 2 citations dans Numdam

DOI : 10.24033/bsmf.2448

Classification : 37E05, 37D25, 37D45, 37C45
Keywords: quadratic map, Jakobson's theorem, Hausdorff dimension, Markov partition, Bernoulli map, induced expansion, absolutely continuous invariant probability measure
Mot clés : application quadratique, théorème de Jakobson, dimension de Hausdorff, partition de Markov, application de Bernoulli, expansion induite, mesure de probabilité invariante absolument continue

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     title = {Dimension of weakly expanding points for quadratic maps},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {399--420},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {131},
     number = {3},
     year = {2003},
     doi = {10.24033/bsmf.2448},
     mrnumber = {2017145},
     zbl = {1071.37028},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2448/}
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Senti, Samuel. Dimension of weakly expanding points for quadratic maps. Bulletin de la Société Mathématique de France, Tome 131 (2003) no. 3, pp. 399-420. doi : 10.24033/bsmf.2448. https://www.numdam.org/articles/10.24033/bsmf.2448/

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