scholarly journals Affine embeddings of Cantor sets on the line

2018 ◽  
Vol 5 (4) ◽  
pp. 339-350 ◽  
Author(s):  
Amir Algom
2018 ◽  
Vol 226 (2) ◽  
pp. 805-826 ◽  
Author(s):  
De-Jun Feng ◽  
Ying Xiong

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1750003 ◽  
Author(s):  
YING ZENG

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].


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

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

Author(s):  
Alberto Takase

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.


2020 ◽  
Vol 379 (1) ◽  
pp. 103-143
Author(s):  
Oleg Kozlovski ◽  
Sebastian van Strien

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.


2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto

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.


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