- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 22:02:47
3
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JFG&vol=2&iss=1&update_since=2024-03-28
Journal of Fractal Geometry
J. Fractal Geom.
JFG
2308-1309
2308-1317
Measure and integration
General
10.4171/JFG
http://www.ems-ph.org/doi/10.4171/JFG
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
2
2015
1
Spectral gaps of almost Mathieu operators in the exponential regime
Wencai
Liu
Fudan University, SHANGHAI, CHINA
Xiaoping
Yuan
Fudan University, SHANGHAI, CHINA
Almost Mathieu operator, Aubry duality, Ten Martini Problem, reducibility
For almost Mathieu operator \[(H_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+2\lambda \cos2\pi(\theta+n\alpha)u_n,\] the dry version of the Ten Martini Problem predicts that the spectrum $\Sigma_{\lambda,\alpha}$ of $ H_{\lambda,\alpha,\theta}$ has all gaps open for all $\lambda\neq 0$ and $ \alpha \in \mathbb{R}\backslash \mathbb{Q}$. Avila and Jitomirskaya prove that $\Sigma_{\lambda,\alpha}$ has all gaps open for Diophantine $\alpha$ and $0
Ordinary differential equations
Operator theory
1
51
10.4171/JFG/15
http://www.ems-ph.org/doi/10.4171/JFG/15
Lipschitz equivalence of self-similar sets and hyperbolic boundaries II
Guo-Tai
Deng
Central China Normal University, WUHAN, CHINA
Ka-Sing
Lau
The Chinese University of Hong Kong, HONG KONG, CHINA
Jun
Luo
Chongqing University, CHONGQING, CHINA
Self-similar set, augmented tree, hyperbolic boundary, near-isometry, Lipschitz equivalence, rearrangeable matrix
In [13], two of the authors gave a study of Lipschitz equivalence of self-similar sets through the augmented trees, a class of hyperbolic graphs introduced by Kaimanovich in [9] and developed by Lau and Wang [10]. In this paper, we continue such investigation. We remove a major assumption in the main theorem in [13] by using a new notion of quasi-rearrangeable matrix, and show that the hyperbolic boundary of any simple augmented tree is Lipschitz equivalent to a Cantor-type set. We then apply this result to consider the Lipschitz equivalence of certain totally disconnected self-similar sets as well as their unions.
Measure and integration
Combinatorics
53
79
10.4171/JFG/16
http://www.ems-ph.org/doi/10.4171/JFG/16
Approximation using hidden variable fractal interpolation function
Arya
Chand
Indian Institute of Technology Madras, CHENNAI, INDIA
Saurabh
Katiyar
Indian Institute of Technology Madras, CHENNAI, INDIA
Puthan
Viswanathan
Indian Institute of Technology Madras, CHENNAI, INDIA
Fractal operator, fractal cubic spline, hidden variable fractal interpolation function, convex optimization, sensitivity analysis, positivity, parameter identification
The notion of hidden variable fractal interpolation provides a method to approximate functions that are self-referential or non-self-referential, and consequently allows great flexibility and diversity for the fractal modeling problem. The current article intends to apply hidden variable fractal interpolation to associate a class of $\mathbb R^2$-valued continuous fractal functions with a prescribed continuous function. Suitable values of the parameters are identified so that the fractal functions retain positivity and regularity of the germ function. As an application of the developed theory, we obtain positive $\mathcal C^1$-cubic spline hidden variable fractal interpolation functions corresponding to a prescribed set of positive data, thus initiating a new approach to shape preserving approximation via hidden variable fractal function. Depending on the values of the parameters, these positive interpolants can reflect the self-referentiality or non-self-referentiality of the original data defining function and fractality of its derivative. Therefore, the present scheme outperforms the traditional nonrecursive positivity preserving $\mathcal C^1$-cubic spline interpolation scheme and its fractal extension studied recently in the literature.
Measure and integration
Approximations and expansions
81
114
10.4171/JFG/17
http://www.ems-ph.org/doi/10.4171/JFG/17