Dimension of weakly expanding points for quadratic maps

Samuel Senti

Bulletin de la Société Mathématique de France (2003)

  • Volume: 131, Issue: 3, page 399-420
  • ISSN: 0037-9484

Abstract

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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 ∈ 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 ≤ | 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 ∈ ℛ (Jakobson’s Theorem).

How to cite

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Senti, Samuel. "Dimension of weakly expanding points for quadratic maps." Bulletin de la Société Mathématique de France 131.3 (2003): 399-420. <http://eudml.org/doc/272388>.

@article{Senti2003,
abstract = {For the real quadratic map $P_a(x)=x^2+a$ and a given $\epsilon &gt;0$ 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 $\epsilon &gt;0$ 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 -

References

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