Dimension of weakly expanding points for quadratic maps

@article{Senti2003,
abstract = {For the real quadratic map $P_a(x)=x^2+a$ and a given $\epsilon >0$ 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 $B(0, \epsilon )\subseteq P_a^n(J)$ for some $n\in \mathbb \{N\}$. The $\epsilon $-weakly expanding set is the set of points which do not have good expansion properties. Let $\alpha $ denote the negative fixed point and $M$ the first return time of the critical orbit to $[\alpha , -\alpha ]$. We show there is a set $\{\mathcal \{R\}\}$ of parameters with positive Lebesgue measure for which the Hausdorff dimension of the $\epsilon $-weakly expanding set is bounded above and below by $\{\log _2\{M\}\}/\{M\}+\{\mathcal \{O\}\}(\{\log _2\{\log _2\{M\}\}\}/\{M\})$ for $\epsilon $ close to $|\alpha |$. For arbitrary $\epsilon \le |\alpha |$ the dimension is of the order of $\{\mathcal \{O\}\}(\{\log _2\{|\log _2\{\epsilon \}|\}\}/\{|\log _2\{\epsilon \}|\}).$ Constants depend only on $M$. The Folklore Theorem then implies the existence of an absolutely continuous invariant probability measure for $P_a$ with $a\in \{\mathcal \{R\}\}$ (Jakobson’s Theorem).},
author = {Senti, Samuel},
journal = {Bulletin de la Société Mathématique de France},
keywords = {quadratic map; Jakobson’s theorem; Hausdorff dimension; Markov partition; Bernoulli map; induced expansion; absolutely continuous invariant probability measure},
language = {eng},
number = {3},
pages = {399-420},
publisher = {Société mathématique de France},
title = {Dimension of weakly expanding points for quadratic maps},
url = {http://eudml.org/doc/272388},
volume = {131},
year = {2003},
}

TY - JOUR
AU - Senti, Samuel
TI - Dimension of weakly expanding points for quadratic maps
JO - Bulletin de la Société Mathématique de France
PY - 2003
PB - Société mathématique de France
VL - 131
IS - 3
SP - 399
EP - 420
AB - For the real quadratic map $P_a(x)=x^2+a$ and a given $\epsilon >0$ 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 $B(0, \epsilon )\subseteq P_a^n(J)$ for some $n\in \mathbb {N}$. The $\epsilon $-weakly expanding set is the set of points which do not have good expansion properties. Let $\alpha $ denote the negative fixed point and $M$ the first return time of the critical orbit to $[\alpha , -\alpha ]$. We show there is a set ${\mathcal {R}}$ of parameters with positive Lebesgue measure for which the Hausdorff dimension of the $\epsilon $-weakly expanding set is bounded above and below by ${\log _2{M}}/{M}+{\mathcal {O}}({\log _2{\log _2{M}}}/{M})$ for $\epsilon $ close to $|\alpha |$. For arbitrary $\epsilon \le |\alpha |$ the dimension is of the order of ${\mathcal {O}}({\log _2{|\log _2{\epsilon }|}}/{|\log _2{\epsilon }|}).$ Constants depend only on $M$. The Folklore Theorem then implies the existence of an absolutely continuous invariant probability measure for $P_a$ with $a\in {\mathcal {R}}$ (Jakobson’s Theorem).
LA - eng
KW - quadratic map; Jakobson’s theorem; Hausdorff dimension; Markov partition; Bernoulli map; induced expansion; absolutely continuous invariant probability measure
UR - http://eudml.org/doc/272388
ER -