scholarly journals Affine embeddings of Cantor sets on the line

2018 ◽  
Vol 5 (4) ◽  
pp. 339-350 ◽  
Author(s):  
Amir Algom
Keyword(s):  
Cantor Sets ◽  
Affine Embeddings

Affine embeddings of Cantor sets and dimension of αβ-sets

2018 ◽  
Vol 226 (2) ◽  
pp. 805-826 ◽  
Author(s):  
De-Jun Feng ◽  
Ying Xiong
Keyword(s):  
Cantor Sets ◽  
Affine Embeddings

SELF-SIMILAR SUBSETS OF SYMMETRIC CANTOR SETS

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1750003 ◽  
Author(s):  
YING ZENG
Keyword(s):  
Cantor Set ◽  
Cantor Sets ◽  
Self Similar ◽  
Affine Embeddings

This paper concerns the affine embeddings of general symmetric Cantor sets. Under certain condition, we show that if a self-similar set [Formula: see text] can be affinely embedded into a symmetric Cantor set [Formula: see text], then their contractions are rationally commensurable. Our result supports Conjecture 1.2 in [D. J. Feng, W. Huang and H. Rao, Affine embeddings and intersections of Cantor sets, J. Math. Pures Appl. 102 (2014) 1062–1079].

scholarly journals Affine embeddings and intersections of Cantor sets

2014 ◽  
Vol 102 (6) ◽  
pp. 1062-1079 ◽  
Author(s):  
De-Jun Feng ◽  
Wen Huang ◽  
Hui Rao
Keyword(s):  
Cantor Sets ◽  
Affine Embeddings

Symmetric Porosity and Symmetric Cantor Sets

1991 ◽  
Vol 17 (1) ◽  
pp. 19
Author(s):  
Evans
Keyword(s):  
Cantor Sets

scholarly journals On the Spectra of Separable 2D Almost Mathieu Operators

2021 ◽  
Author(s):  
Alberto Takase
Keyword(s):  
Positive Measure ◽  
Schrödinger Operators ◽  
Cantor Sets ◽  
Schrodinger Operators ◽  
Discrete Schrödinger Operators ◽  
Mathieu Operator ◽  
Almost Mathieu Operator

AbstractWe consider separable 2D discrete Schrödinger operators generated by 1D almost Mathieu operators. For fixed Diophantine frequencies, we prove that for sufficiently small couplings the spectrum must be an interval. This complements a result by J. Bourgain establishing that for fixed couplings the spectrum has gaps for some (positive measure) Diophantine frequencies. Our result generalizes to separable multidimensional discrete Schrödinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine. The proof is based on the study of the thickness of the spectrum of the almost Mathieu operator and utilizes the Newhouse Gap Lemma on sums of Cantor sets.

scholarly journals Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws

2020 ◽  
Vol 379 (1) ◽  
pp. 103-143
Author(s):  
Oleg Kozlovski ◽  
Sebastian van Strien
Keyword(s):  
Scaling Laws ◽  
Scaling Law ◽  
Period Doubling ◽  
Cantor Sets ◽  
Unimodal Maps ◽  
Renormalization Theory

Abstract We consider a family of strongly-asymmetric unimodal maps $$\{f_t\}_{t\in [0,1]}$$ { f t } t ∈ [ 0 , 1 ] of the form $$f_t=t\cdot f$$ f t = t · f where $$f:[0,1]\rightarrow [0,1]$$ f : [ 0 , 1 ] → [ 0 , 1 ] is unimodal, $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 , $$f(c)=1$$ f ( c ) = 1 is of the form and $$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)&{} \text{ for } x<c, \\ 1-K_+|x-c|^\beta + o(|x-c|^\beta ) &{} \text{ for } x>c, \end{array}\right. \end{aligned}$$ f ( x ) = 1 - K - | x - c | + o ( | x - c | ) for x < c , 1 - K + | x - c | β + o ( | x - c | β ) for x > c , where we assume that $$\beta >1$$ β > 1 . We show that such a family contains a Feigenbaum–Coullet–Tresser $$2^\infty $$ 2 ∞ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $$2^\infty $$ 2 ∞ map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum–Coullet–Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results.

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.

Sign in / Sign up

Export Citation Format

Share Document