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.

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.

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.

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 Ratio geometry, rigidity and the scenery process for hyperbolic Cantor sets

1997 ◽  
Vol 17 (3) ◽  
pp. 531-564 ◽  
Author(s):  
TIM BEDFORD ◽  
ALBERT M. FISHER
Keyword(s):  
Conjugacy Class ◽  
Cantor Set ◽  
Cantor Sets ◽  
Rigidity Theorem ◽  
Compact Set ◽  
Banach Manifold ◽  
Hölder Continuous ◽  
Limit Sets ◽  
Smoothness Class

Given a ${\cal C}^{1+\gamma}$ hyperbolic Cantor set $C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a Hölder continuous set-valued function defined on Sullivan's dual Cantor set. We show the limit sets are themselves ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the ${\cal C}^{1+\gamma}$ conjugacy class of $C$. The proof of this leads to the following rigidity theorem: if two ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets are ${\cal C}^1$ conjugate, then the conjugacy (with a different extension) is in fact already ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$. Within one ${\cal C}^{1+\gamma}$ conjugacy class, each smoothness class is a Banach manifold, which is acted on by the semigroup given by rescaling subintervals. Smoothness classes nest down, and contained in the intersection of them all is a compact set which is the attractor for the semigroup: the collection of limit sets. Convergence is exponentially fast, in the ${\cal C}^1$ norm.

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].

Limit Sets of Typical Homeomorphisms

2012 ◽  
Vol 55 (2) ◽  
pp. 225-232 ◽  
Author(s):  
Nilson C. Bernardes
Keyword(s):  
Hausdorff Dimension ◽  
Borel Measure ◽  
Cantor Set ◽  
Limit Set ◽  
Positive Borel Measure ◽  
Dense Orbit ◽  
Limit Sets ◽  
Dimension Zero

AbstractGiven an integer n ≥ 3, a metrizable compact topological n-manifold X with boundary, and a finite positive Borel measure μ on X, we prove that for the typical homeomorphism f : X → X, it is true that for μ-almost every point x in X the limit set ω( f, x) is a Cantor set of Hausdorff dimension zero, each point of ω(f, x) has a dense orbit in ω(f, x), f is non-sensitive at each point of ω(f, x), and the function a → ω(f, a) is continuous at x.

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.

Superior Cantor Sets and Superior Devil Staircases

2010 ◽  
Vol 1 (1) ◽  
pp. 78-84 ◽  
Author(s):  
Mamta Rani ◽  
Sanjeev Kumar Prasad
Keyword(s):  
Dynamical Systems ◽  
Real World ◽  
Nonlinear Dynamical Systems ◽  
Cantor Set ◽  
Cantor Sets ◽  
Strange Attractors ◽  
Nonlinear Dynamical ◽  
The Universe ◽  
Real World Problems

Mandelbrot, in 1975, coined the term fractal and included Cantor set as a classical example of fractals. The Cantor set has wide applications in real world problems from strange attractors of nonlinear dynamical systems to the distribution of galaxies in the universe (Schroder, 1990). In this article, we obtain superior Cantor sets and present them graphically by superior devil’s staircases. Further, based on their method of generation, we put them into two categories.

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.

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