Dimension of weakly expanding points for quadratic maps
Bulletin de la Société Mathématique de France, Volume 131 (2003) no. 3, p. 399-420

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)P a n (J) for some 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 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|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 (Jakobson’s Theorem).

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)P a n (J) pour un 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 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|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 (théorème de Jakobson).

DOI : https://doi.org/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
@article{BSMF_2003__131_3_399_0,
     author = {Senti, Samuel},
     title = {Dimension of weakly expanding points for quadratic maps},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {131},
     number = {3},
     year = {2003},
     pages = {399-420},
     doi = {10.24033/bsmf.2448},
     zbl = {1071.37028},
     mrnumber = {2017145},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2003__131_3_399_0}
}
Senti, Samuel. Dimension of weakly expanding points for quadratic maps. Bulletin de la Société Mathématique de France, Volume 131 (2003) no. 3, pp. 399-420. doi : 10.24033/bsmf.2448. http://www.numdam.org/item/BSMF_2003__131_3_399_0/

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