### Abstract

For a positive measure set of non-uniformly expanding quadratic maps on the interval we effect a multifractal formalism, i.e., decompose the phase space into level sets of time averages of a given continuous function and consider the associated Birkhoff spectrum which encodes this decomposition. We derive a formula which relates the Hausdorff dimension of level sets to entropies and Lyapunov exponents of invariant probability measures, and then use this formula to show that the spectrum is continuous. In order to estimate the Hausdorff dimension from above, one has to 'see' sufficiently many points. To this end, we construct a family of towers. Using these towers we establish a large deviation principle of empirical distributions, with Lebesgue as a reference measure.

Original language | English |
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Pages (from-to) | 1116-1141 |

Number of pages | 26 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 34 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2014 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Ergodic Theory and Dynamical Systems*,

*34*(4), 1116-1141. https://doi.org/10.1017/etds.2012.188

**Multifractal formalism for Benedicks-Carleson quadratic maps.** / Chung, Yong Moo; Takahasi, Hiroki.

Research output: Contribution to journal › Article

*Ergodic Theory and Dynamical Systems*, vol. 34, no. 4, pp. 1116-1141. https://doi.org/10.1017/etds.2012.188

}

TY - JOUR

T1 - Multifractal formalism for Benedicks-Carleson quadratic maps

AU - Chung, Yong Moo

AU - Takahasi, Hiroki

PY - 2014

Y1 - 2014

N2 - For a positive measure set of non-uniformly expanding quadratic maps on the interval we effect a multifractal formalism, i.e., decompose the phase space into level sets of time averages of a given continuous function and consider the associated Birkhoff spectrum which encodes this decomposition. We derive a formula which relates the Hausdorff dimension of level sets to entropies and Lyapunov exponents of invariant probability measures, and then use this formula to show that the spectrum is continuous. In order to estimate the Hausdorff dimension from above, one has to 'see' sufficiently many points. To this end, we construct a family of towers. Using these towers we establish a large deviation principle of empirical distributions, with Lebesgue as a reference measure.

AB - For a positive measure set of non-uniformly expanding quadratic maps on the interval we effect a multifractal formalism, i.e., decompose the phase space into level sets of time averages of a given continuous function and consider the associated Birkhoff spectrum which encodes this decomposition. We derive a formula which relates the Hausdorff dimension of level sets to entropies and Lyapunov exponents of invariant probability measures, and then use this formula to show that the spectrum is continuous. In order to estimate the Hausdorff dimension from above, one has to 'see' sufficiently many points. To this end, we construct a family of towers. Using these towers we establish a large deviation principle of empirical distributions, with Lebesgue as a reference measure.

UR - http://www.scopus.com/inward/record.url?scp=84903946659&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84903946659&partnerID=8YFLogxK

U2 - 10.1017/etds.2012.188

DO - 10.1017/etds.2012.188

M3 - Article

AN - SCOPUS:84903946659

VL - 34

SP - 1116

EP - 1141

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 4

ER -