scholarly journals Hyperbolization of the limit sets of some geometric constructions

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1535-1544
Author(s):  
Zhanqi Zhang ◽  
Yingqing Xiao
Keyword(s):  
Hyperbolic Space ◽  
Geometric Constructions ◽  
Limit Sets ◽  
Fractal Sets ◽  
Fractal Set ◽  
New Class ◽  
Gromov Hyperbolic ◽  
Hyperbolic Boundary

Inspired by the construction of Sierpi?ski carpets, we introduce a new class of fractal sets. For a such fractal set K, we construct a Gromov hyperbolic space X (which is also a strongly hyperbolic space) and show that K is isometric to the Gromov hyperbolic boundary of X. Moreover, under some conditions, we show that Con(K) and X are roughly isometric, where Con(K) is the hyperbolic cone of K.

scholarly journals Uniform cantor sets as hyperbolic boundaries

Filomat ◽  
2014 ◽  
Vol 28 (8) ◽  
pp. 1737-1745 ◽  
Author(s):  
Yingqing Xiao ◽  
Junping Gu
Keyword(s):  
Hyperbolic Space ◽  
Cantor Set ◽  
Cantor Sets ◽  
Gromov Hyperbolic ◽  
Hyperbolic Boundary

In this paper, we prove that the Gromov hyperbolic space (?,h) which was introduced by Z. Ibragimov and J. Simanyi in [3] is an asymptotically PT??1 space and extend the methods of [3] to the case of uniform Cantor sets, show that the uniform Cantor set is isometric to the Gromov hyperbolic boundary at infinity of some asymptotically PT-1 space.

The Bishop–Jones relation and Hausdorff geometry of convex-cobounded limit sets in infinite-dimensional hyperbolic space

2016 ◽  
Vol 16 (05) ◽  
pp. 1650018 ◽  
Author(s):  
Tushar Das ◽  
Bernd O. Stratmann ◽  
Mariusz Urbański
Keyword(s):  
Hyperbolic Space ◽  
Limit Sets ◽  
Multiplicative Constant ◽  
Infinite Dimensional ◽  
Mass Redistribution ◽  
Isometric Actions

We generalize the mass redistribution principle and apply it to prove the Bishop–Jones relation for limit sets of metrically proper isometric actions on real infinite-dimensional hyperbolic space. We also show that the Hausdorff and packing measures on the limit sets of convex-cobounded groups are finite and positive and coincide with the conformal Patterson measure, up to a multiplicative constant.

scholarly journals Normalizer, divergence type, and Patterson measure for discrete groups of the Gromov hyperbolic space

2020 ◽  
Vol 14 (2) ◽  
pp. 369-411
Author(s):  
Katsuhiko Matsuzaki ◽  
Yasuhiro Yabuki ◽  
Johannes Jaerisch
Keyword(s):  
Hyperbolic Space ◽  
Discrete Groups ◽  
Gromov Hyperbolic ◽  
Divergence Type

AUTOMATA ON FRACTAL SETS OBSERVED IN HYBRID DYNAMICAL SYSTEMS

2008 ◽  
Vol 18 (12) ◽  
pp. 3665-3678 ◽  
Author(s):  
JUN NISHIKAWA ◽  
KAZUTOSHI GOHARA
Keyword(s):  
Dynamical System ◽  
Information Processing ◽  
Dynamical Systems ◽  
Feedback System ◽  
The Other ◽  
Hybrid Dynamical Systems ◽  
Discrete Dynamics ◽  
Hybrid Dynamical System ◽  
Fractal Sets ◽  
Fractal Set

We studied a hybrid dynamical system composed of a higher module with discrete dynamics and a lower module with continuous dynamics. Two typical examples of this system were investigated from the viewpoint of dynamical systems. One example is a nonfeedback system whose higher module stochastically switches inputs to the lower module. The dynamics was characterized by attractive and invariant fractal sets with hierarchical clusters addressed by input sequences. The other example is a feedback system whose higher module switches in response to the states of the lower module at regular intervals. This system converged into various switching attractors that correspond to infinite switching manifolds, which define each feedback control rule at the switching point. We showed that the switching attractors in the feedback system are subsets of the fractal sets in the nonfeedback system. The feedback system can be considered an automaton that generates various sequences from the fractal set by choosing the typical switching manifold. We can control this system by adjusting the switching interval to determine the fractal set as a constraint and by adjusting the switching manifold to select the automaton from the fractal set. This mechanism might be the key to developing information processing that is neither too soft nor too rigid.

CERTAIN INTEGRAL INEQUALITIES CONSIDERING GENERALIZED m-CONVEXITY ON FRACTAL SETS AND THEIR APPLICATIONS

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950117 ◽  
Author(s):  
TINGSONG DU ◽  
HAO WANG ◽  
MUHAMMAD ADIL KHAN ◽  
YAO ZHANG
Keyword(s):  
Probability Density ◽  
Fractional Integrals ◽  
The Real ◽  
Text Type ◽  
Fractal Sets ◽  
Convex Mappings ◽  
Fractal Set ◽  
Special Means ◽  
Real Linear ◽  
Generalized Probability

First, we introduce a generalized [Formula: see text]-convexity concept defined on the real linear fractal set [Formula: see text] [Formula: see text] and discuss the relation between generalized [Formula: see text]-convexity and [Formula: see text]-convexity. Second, we present several important properties of the generalized [Formula: see text]-convex mappings. Meanwhile, via local fractional integrals, we also derive certain estimation-type results on generalizations of Hadamard-type, Fejér-type and Simpson-type inequalities. As applications related to local fractional integrals, we construct several inequalities for generalized probability density mappings and [Formula: see text]-type special means.

scholarly journals The laplacian for domains in hyperbolic space and limit sets of Kleinian groups

1985 ◽  
Vol 155 (0) ◽  
pp. 173-241 ◽  
Author(s):  
R. S. Phillips ◽  
P. Sarnak
Keyword(s):  
Hyperbolic Space ◽  
Kleinian Groups ◽  
Limit Sets
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