The Hausdorff dimension of level sets described by Erdös–Rényi average

Let [Formula: see text] denote the largest digit of the first [Formula: see text] terms in the Lüroth expansion of [Formula: see text]. Shen, Yu and Zhou, A note on the largest digits in Luroth expansion, Int. J. Number Theory 10 (2014) 1015–1023 considered the level sets [Formula: see text] and proved that each [Formula: see text] has full Hausdorff dimension. In this paper, we investigate the Hausdorff dimension of the following refined exceptional set: [Formula: see text] and show that [Formula: see text] has full Hausdorff dimension for each pair [Formula: see text] with [Formula: see text]. Combining the two results, [Formula: see text] can be decomposed into the disjoint union of uncountably many sets with full Hausdorff dimension.

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A classification of aperiodic order via spectral metrics and Jarník sets

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.7 ◽

2018 ◽

Vol 39 (11) ◽

pp. 3031-3065 ◽

Cited By ~ 1

Author(s):

MAIK GRÖGER ◽

MARC KESSEBÖHMER ◽

ARNE MOSBACH ◽

TONY SAMUEL ◽

MALTE STEFFENS

Keyword(s):

Hausdorff Dimension ◽

Level Sets ◽

Continued Fraction ◽

Regularity Condition ◽

Hölder Regularity ◽

Aperiodic Order ◽

Holder Regularity

Given an$\unicode[STIX]{x1D6FC}>1$and a$\unicode[STIX]{x1D703}$with unbounded continued fraction entries, we characterize new relations between Sturmian subshifts with slope$\unicode[STIX]{x1D703}$with respect to (i) an$\unicode[STIX]{x1D6FC}$-Hölder regularity condition of a spectral metric, (ii) level sets defined in terms of the Diophantine properties of$\unicode[STIX]{x1D703}$, and (iii) complexity notions which we call$\unicode[STIX]{x1D6FC}$-repetitiveness,$\unicode[STIX]{x1D6FC}$-repulsiveness and$\unicode[STIX]{x1D6FC}$-finiteness—generalizations of the properties known as linear repetitiveness, repulsiveness and power freeness, respectively. We show that the level sets relate naturally to (exact) Jarník sets and prove that their Hausdorff dimension is$2/(\unicode[STIX]{x1D6FC}+1)$.

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The Hausdorff Dimension of the Level Sets for a Fractional Brownian Sheet

Stochastic Analysis and Applications ◽

10.1081/sap-200029491 ◽

2004 ◽

Vol 22 (6) ◽

pp. 1511-1523

Author(s):

Rongmao Zhang ◽

Zhengyan Lin

Keyword(s):

Hausdorff Dimension ◽

Level Sets ◽

Brownian Sheet ◽

Fractional Brownian Sheet

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The Lyapunov spectrum of some parabolic systems

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385708080462 ◽

2009 ◽

Vol 29 (3) ◽

pp. 919-940 ◽

Cited By ~ 30

Author(s):

KATRIN GELFERT ◽

MICHAŁ RAMS

Keyword(s):

Lyapunov Exponent ◽

Hausdorff Dimension ◽

Lyapunov Exponents ◽

Topological Entropy ◽

Level Set ◽

Level Sets ◽

Parabolic Systems ◽

Lyapunov Spectrum ◽

Interval Maps ◽

Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

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The Hausdorff Dimension of the Generalized Level Sets of Takagi's Function

Real Analysis Exchange ◽

10.14321/realanalexch.38.2.0421 ◽

2013 ◽

Vol 38 (2) ◽

pp. 421

Author(s):

Enrique de Amo ◽

Manuel Díaz Carrillo ◽

Juan Fernández Sánchez

Keyword(s):

Hausdorff Dimension ◽

Level Sets ◽

Takagi’S Function

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Multifractal formalism for Benedicks–Carleson quadratic maps

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2012.188 ◽

2013 ◽

Vol 34 (4) ◽

pp. 1116-1141 ◽

Cited By ~ 7

Author(s):

YONG MOO CHUNG ◽

HIROKI TAKAHASI

Keyword(s):

Hausdorff Dimension ◽

Level Sets ◽

Large Deviation Principle ◽

Large Deviation ◽

Probability Measures ◽

Multifractal Formalism ◽

Quadratic Maps ◽

Empirical Distributions ◽

Reference Measure ◽

Time Averages

AbstractFor 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.

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