On the Hausdorff dimension of general Cantor sets

1965 ◽  
Vol 61 (3) ◽  
pp. 679-694 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Hausdorff Dimension ◽  
Cantor Set ◽  
Cantor Sets ◽  
Local Dimension ◽  
Underlying Space

Introduction and notation. In this paper a generalization of the Cantor set is discussed. Upper and lower estimates of the Hausdorff dimension of such a set are obtained and, in particular, it is shown that the Hausdorff dimension is always positive and less than that of the underlying space. The concept of local dimension at a point is introduced and studied as a function of that point.

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

A REMARK ON SELF-SIMILAR SUBSETS OF UNIFORM CANTOR SETS WITH SMALL HAUSDORFF DIMENSION

Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050057
Author(s):  
HUI RAO ◽  
ZHI-YING WEN ◽  
YING ZENG
Keyword(s):  
Hausdorff Dimension ◽  
Difficult Problem ◽  
Cantor Set ◽  
Cantor Sets ◽  
Symbolic Space ◽  
Self Similar

Recently there are several works devoted to the study of self-similar subsets of a given self-similar set, which turns out to be a difficult problem. Let [Formula: see text] be an integer and let [Formula: see text]. Let [Formula: see text] be the uniform Cantor set defined by the following set equation: [Formula: see text] We show that for any [Formula: see text], [Formula: see text] and [Formula: see text] essentially have the same self-similar subsets. Precisely, [Formula: see text] is a self-similar subset of [Formula: see text] if and only if [Formula: see text] is a self-similar subset of [Formula: see text], where [Formula: see text] (similarly [Formula: see text]) is the coding map from the symbolic space [Formula: see text] to [Formula: see text].

Maximal Operators and Cantor Sets

2000 ◽  
Vol 43 (3) ◽  
pp. 330-342 ◽  
Author(s):  
Kathryn E. Hare
Keyword(s):  
Hausdorff Dimension ◽  
Cantor Set ◽  
Cantor Sets ◽  
Maximal Operators

AbstractWe consider maximal operators in the plane, defined by Cantor sets of directions, and show such operators are not bounded on L2 if the Cantor set has positive Hausdorff dimension.

scholarly journals INTERSECTIONS OF CERTAIN DELETED DIGITS SETS

Fractals ◽  
2012 ◽  
Vol 20 (01) ◽  
pp. 105-115 ◽  
Author(s):  
STEEN PEDERSEN ◽  
JASON D. PHILLIPS
Keyword(s):  
Hausdorff Dimension ◽  
Cantor Set ◽  
Cantor Sets ◽  
Digit Set

We consider some properties of the intersection of deleted digits Cantor sets with their translates. We investigate conditions on the set of digits such that, for any t between zero and the dimension of the deleted digits Cantor set itself, the set of translations such that the intersection has that Hausdorff dimension equal to t is dense in the set F of translations such that the intersection is non-empty. We make some simple observations regarding properties of the set F, in particular, we characterize when F is an interval, in terms of conditions on the digit set.

scholarly journals Basmajian-type identities and Hausdorff dimension of limit sets

2017 ◽  
Vol 38 (6) ◽  
pp. 2224-2244
Author(s):  
YAN MARY HE
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Cantor Set ◽  
Cantor Sets ◽  
Complex Dynamical Systems ◽  
Limit Sets ◽  
One Dimensional ◽  
Type Series ◽  
Trivial Monodromy

In this paper, we study Basmajian-type series identities on holomorphic families of Cantor sets associated to one-dimensional complex dynamical systems. We show that the series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is strictly less than one. Throughout the domain of convergence, these identities can be analytically continued and they exhibit non-trivial monodromy.

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].

Quasisymmetrically Minimal Moran Sets

2013 ◽  
Vol 56 (2) ◽  
pp. 292-305 ◽  
Author(s):  
Mei-Feng Dai
Keyword(s):  
Hausdorff Dimension ◽  
Cantor Sets ◽  
Equal Length

AbstractM. Hu and S. Wen considered quasisymmetrically minimal uniform Cantor sets of Hausdorff dimension 1, where at the k-th set one removes from each interval I a certain number nk of open subintervals of length ck|I|, leaving (nk + 1) closed subintervals of equal length. Quasisymmetrically Moran sets of Hausdorff dimension 1 considered in the paper are more general than uniform Cantor sets in that neither the open subintervals nor the closed subintervals are required to be of equal length.

scholarly journals On the structure of the family of Cherry fields on the torus

1985 ◽  
Vol 5 (1) ◽  
pp. 27-46 ◽  
Author(s):  
Colin Boyd
Keyword(s):  
Vector Field ◽  
Hausdorff Dimension ◽  
Rotation Number ◽  
Vector Fields ◽  
Cantor Set ◽  
Irrational Rotation ◽  
Codimension One ◽  
The Family ◽  
Irrational Rotation Number

AbstractA class of vector fields on the 2-torus, which includes Cherry fields, is studied. Natural paths through this class are defined and it is shown that the parameters for which the vector field is unstable is the closure ofhas irrational rotation number}, where ƒ is a certain map of the circle andRtis rotation throught. This is shown to be a Cantor set of zero Hausdorff dimension. The Cherry fields are shown to form a family of codimension one submanifolds of the set of vector fields. The natural paths are shown to be stable paths.

FOX-ARTIN ARC AND CONSTRUCTING KLEINIAN GROUPS ACTING ON S3 WITH LIMIT SET A WILD CANTOR SET

1995 ◽  
Vol 06 (01) ◽  
pp. 19-32 ◽  
Author(s):  
NIKOLAY GUSEVSKII ◽  
HELEN KLIMENKO
Keyword(s):  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
Limit Set ◽  
Limit Sets ◽  
Geometrically Finite ◽  
Wild Cantor Set ◽  
Wild Cantor Sets

We construct purely loxodromic, geometrically finite, free Kleinian groups acting on S3 whose limit sets are wild Cantor sets. Our construction is closely related to the construction of the wild Fox–Artin arc.

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