scholarly journals Dynamics of iterated function systems on the circle close to rotations

2014 ◽  
Vol 35 (5) ◽  
pp. 1345-1368 ◽  
Author(s):  
PABLO G. BARRIENTOS ◽  
ARTEM RAIBEKAS
Keyword(s):  
Type Theorem ◽  
Iterated Function Systems ◽  
Cantor Sets ◽  
Limit Set ◽  
Iterated Function ◽  
Circle Diffeomorphisms

We study the dynamics of iterated function systems generated by a pair of circle diffeomorphisms close to rotations in the $C^{1+\text{bv}}$-topology. We characterize the obstruction to minimality and describe the limit set. In particular, there are no invariant minimal Cantor sets, which can be seen as a Denjoy/Duminy type theorem for iterated systems on the circle.

FRACTAL GEOMETRY OF RESTRICTED SETS OF CIRCLE INVERSIONS

Fractals ◽  
1995 ◽  
Vol 03 (04) ◽  
pp. 689-699 ◽  
Author(s):  
COLLEEN CLANCY ◽  
MICHAEL FRAME
Keyword(s):  
Fractal Geometry ◽  
Graphical Representation ◽  
Iterated Function Systems ◽  
Visual Presentation ◽  
Limit Set ◽  
Limit Sets ◽  
Iterated Function ◽  
Grammatical Complexity

For intersecting circles, we propose a modification of the limit set generated by inversion in circles. This restricted limit set is always a subset of the discs bounded by the generating circles. We give examples of restricted limit sets and show arrangements of generating circles for which the restricted limit set equals the limit set, and also arrangements for which they differ. In addition, we give a visual presentation, based on Iterated Function Systems, of the excluded strings in the restricted limit set. This leads to a graphical representation of the grammatical complexity of the restricted limit set.

scholarly journals Sampling and interpolation of cumulative distribution functions of Cantor sets in [0, 1]

2021 ◽  
Vol 54 (1) ◽  
pp. 85-109
Author(s):  
Allison Byars ◽  
Evan Camrud ◽  
Steven N. Harding ◽  
Sarah McCarty ◽  
Keith Sullivan ◽  
...  
Keyword(s):  
Cumulative Distribution Function ◽  
Invariant Measures ◽  
Borel Probability Measure ◽  
Iterated Function Systems ◽  
Distribution Functions ◽  
Cantor Sets ◽  
Cumulative Distribution ◽  
Sampling Schemes ◽  
Iterated Function ◽  
Borel Probability

Abstract Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures.

GENERALIZED ITERATED FUNCTION SYSTEMS ON HYPERBOLIC NUMBER PLANE

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950045
Author(s):  
GAMALIEL YAFTE TÉLLEZ-SÁNCHEZ ◽  
JUAN BORY-REYES
Keyword(s):  
Fixed Point ◽  
Real Line ◽  
Unique Fixed Point ◽  
Iterated Function Systems ◽  
Cantor Sets ◽  
Affine Transformations ◽  
The Real ◽  
Fractal Sets ◽  
Hyperbolic Number ◽  
Iterated Function

Iterated function systems provide the most fundamental framework to create many fascinating fractal sets. They have been extensively studied when the functions are affine transformations of Euclidean spaces. This paper investigates the iterated function systems consisting of affine transformations of the hyperbolic number plane. We show that the basics results of the classical Hutchinson–Barnsley theory can be carried over to construct fractal sets on hyperbolic number plane as its unique fixed point. We also discuss about the notion of hyperbolic derivative of an hyperbolic-valued function and then we use this notion to get some generalization of cookie-cutter Cantor sets in the real line to the hyperbolic number plane.

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