Multifractal formalism for Benedicks-Carleson quadratic maps

Yong Moo Chung, Hiroki Takahasi

Research output: Contribution to journalArticle

6 Citations (Scopus)

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 languageEnglish
Pages (from-to)1116-1141
Number of pages26
JournalErgodic Theory and Dynamical Systems
Volume34
Issue number4
DOIs
Publication statusPublished - 2014
Externally publishedYes

Fingerprint

Multifractal Formalism
Quadratic Map
Hausdorff Dimension
Level Set
Towers
Decompose
Expanding Maps
Large Deviation Principle
Empirical Distribution
Time-average
Henri Léon Lebésgue
Invariant Measure
Lyapunov Exponent
Probability Measure
Phase Space
Continuous Function
Entropy
Decomposition
Interval
Estimate

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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

In: Ergodic Theory and Dynamical Systems, Vol. 34, No. 4, 2014, p. 1116-1141.

Research output: Contribution to journalArticle

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