scholarly journals A classification of aperiodic order via spectral metrics and Jarník sets

2018 ◽  
Vol 39 (11) ◽  
pp. 3031-3065 ◽  
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$2/(\unicode[STIX]{x1D6FC}+1)$.

scholarly journals Directional Multifractal Analysis in the Lp Setting

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Mourad Ben Slimane ◽  
Ines Ben Omrane ◽  
Moez Ben Abid ◽  
Borhen Halouani ◽  
Farouq Alshormani
Keyword(s):  
Hausdorff Dimension ◽  
Multifractal Analysis ◽  
Hölder Regularity ◽  
Wavelet Coefficients ◽  
Wide Range ◽  
Bounded Functions ◽  
Set Of Points ◽  
Holder Regularity

The classical Hölder regularity is restricted to locally bounded functions and takes only positive values. The local Lp regularity covers unbounded functions and negative values. Nevertheless, it has the same apparent regularity in all directions. In the present work, we study a recent notion of directional local Lp regularity introduced by Jaffard. We provide its characterization by a supremum of a wide range oriented anisotropic Triebel wavelet coefficients and leaders. In addition, we deduce estimates on the Hausdorff dimension of the set of points where the directional local Lp regularity does not exceed a given value. The obtained results are illustrated by some examples of self-affine cascade functions.

THE DISTAL AND ASYMPTOTIC SETS FOR CONTINUED FRACTIONS

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950139
Author(s):  
WEIBIN LIU ◽  
SHUAILING WANG
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Underlying Space ◽  
And Topology

For continued fraction dynamical system [Formula: see text], we give a classification of the underlying space [Formula: see text] according to the orbit of a given point [Formula: see text]. The sizes of all classes are determined from the viewpoints of measure, Hausdorff dimension and topology. For instance, the Hausdorff dimension of the distal set of [Formula: see text] is one and the Hausdorff dimension of the asymptotic set is either zero or [Formula: see text] according to [Formula: see text] is rational or not.

Hausdorff dimension of level sets in Engel continued fraction

2018 ◽  
Vol 92 (3-4) ◽  
pp. 317-330
Author(s):  
Kunkun Song ◽  
Lulu Fang ◽  
Yuanyang Chang ◽  
Jihua Ma
Keyword(s):  
Hausdorff Dimension ◽  
Level Sets ◽  
Continued Fraction

scholarly journals The local structure of the free boundary in the fractional obstacle problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Matteo Focardi ◽  
Emanuele Spadaro
Keyword(s):  
Hausdorff Dimension ◽  
Free Boundary ◽  
Hausdorff Measure ◽  
Local Structure ◽  
Obstacle Problem ◽  
List Type ◽  
Minkowski Content ◽  
Local Finiteness ◽  
Lower Dimensional

AbstractBuilding upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results:(i)local finiteness of the {(n-1)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\mathcal{H}}^{n-1}}-rectifiability of the free boundary,(iii)classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most {(n-2)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {k\geq 2}, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than {k+1}.

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

2004 ◽  
Vol 04 (01) ◽  
pp. 63-76 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Hausdorff Dimension ◽  
Finite Subset ◽  
Continued Fraction ◽  
Cantor Set ◽  
Computational Method ◽  
Accurate Approximation ◽  
Natural Numbers ◽  
Important Special Case ◽  
The One ◽  
Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

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

2018 ◽  
Vol 458 (1) ◽  
pp. 464-480
Author(s):  
Haibo Chen ◽  
Daoxin Ding ◽  
Xinghuo Long
Keyword(s):  
Hausdorff Dimension ◽  
Level Sets
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