scholarly journals Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension

2012 ◽  
Vol 32 (7) ◽  
pp. 2417-2436 ◽  
Author(s):  
Doug Hensley ◽  
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Cantor Sets ◽  
Analytic Extension ◽  
Transfer Operators

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.

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 Hausdorff dimension of an exceptional set in the theory of continued fractions

Nonlinearity ◽  
2020 ◽  
Vol 33 (6) ◽  
pp. 2615-2639 ◽  
Author(s):  
Ayreena Bakhtawar ◽  
Philip Bos ◽  
Mumtaz Hussain
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Exceptional Set

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 Resonance between Cantor sets

2009 ◽  
Vol 29 (1) ◽  
pp. 201-221 ◽  
Author(s):  
YUVAL PERES ◽  
PABLO SHMERKIN
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Unit Interval ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Scaling Parameter ◽  
Iterated Function ◽  
Self Similar ◽  
Image Position ◽  
Geometric Resonance

AbstractLet Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two intervals. We prove that if log b/log a is irrational, then where dim is Hausdorff dimension. More generally, given two self-similar sets K,K′ in ℝ and a scaling parameter s>0, if the dimension of the arithmetic sum K+sK′ is strictly smaller than dim (K)+dim (K′)≤1 (‘geometric resonance’), then there exists r<1 such that all contraction ratios of the similitudes defining K and K′ are powers of r (‘algebraic resonance’). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

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].

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