scholarly journals Hausdorff dimension of the limit set of conformal iterated function systems with overlaps

2011 ◽  
Vol 139 (08) ◽  
pp. 2767-2767 ◽  
Author(s):  
Eugen Mihailescu ◽  
Mariusz Urbański
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function Systems ◽  
Limit Set ◽  
Iterated Function

Parabolic iterated function systems

2000 ◽  
Vol 20 (5) ◽  
pp. 1423-1447 ◽  
Author(s):  
R. D. MAULDIN ◽  
M. URBAŃSKI
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measures ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Topological Pressure ◽  
Limit Set ◽  
Iterated Function ◽  
Conformal Measures ◽  
Conformal Iterated Function System ◽  
Dynamical Features

In this paper we introduce and explore conformal parabolic iterated function systems. We define and study topological pressure, Perron–Frobenius-type operators, semiconformal and conformal measures and the Hausdorff dimension of the limit set. With every parabolic system we associate an infinite hyperbolic conformal iterated function system and we employ it to study geometric and dynamical features (properly defined invariant measures for example) of the limit set.

scholarly journals Exceptional sets for self-similar fractals in Carnot groups

2010 ◽  
Vol 149 (1) ◽  
pp. 147-172 ◽  
Author(s):  
ZOLTÁN M. BALOGH ◽  
RETO BERGER ◽  
ROBERTO MONTI ◽  
JEREMY T. TYSON
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Iterated Function Systems ◽  
Carnot Groups ◽  
Exceptional Set ◽  
Carathéodory Metric ◽  
Exceptional Sets ◽  
Iterated Function ◽  
Self Similar ◽  
Similarity Dimension

AbstractWe consider self-similar iterated function systems in the sub-Riemannian setting of Carnot groups. We estimate the Hausdorff dimension of the exceptional set of translation parameters for which the Hausdorff dimension in terms of the Carnot–Carathéodory metric is strictly less than the similarity dimension. This extends a recent result of Falconer and Miao from Euclidean space to Carnot groups.

scholarly journals The quasi-Assouad dimension of stochastically self-similar sets

2019 ◽  
Vol 150 (1) ◽  
pp. 261-275 ◽  
Author(s):  
Sascha Troscheit
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function Systems ◽  
Random Sets ◽  
Open Set Condition ◽  
Fractal Percolation ◽  
Open Set ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar

AbstractThe class of stochastically self-similar sets contains many famous examples of random sets, for example, Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and V -variable sets and the removal of overlap conditions.

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.

THE BOUNDARY OF THE ATTRACTOR OF A RECURRENT ITERATED FUNCTION SYSTEM

Fractals ◽  
2002 ◽  
Vol 10 (01) ◽  
pp. 77-89 ◽  
Author(s):  
F. M. DEKKING ◽  
P. VAN DER WAL
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Iterated Function

We prove for a subclass of recurrent iterated function systems (also called graph-directed iterated function systems) that the boundary of their attractor is again the attractor of a recurrent IFS. Our method is constructive and permits computation of the Hausdorff dimension of the attractor and its boundary.

SHRINKING TARGET PROBLEM FOR RANDOM IFS

Fractals ◽  
2018 ◽  
Vol 26 (06) ◽  
pp. 1850085
Author(s):  
ZHIHUI YUAN
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Main Idea ◽  
Iterated Function Systems ◽  
Target Problem ◽  
Iterated Function ◽  
Poincare Recurrence ◽  
Poincaré Recurrence

We describe the shrinking target problem for random iterated function systems which are semi-conjugate to random subshifts. We get the Hausdorff dimension of the set based on shrinking target problems with given targets. The main idea is an extension of ubiquity theorem which plays an important role to get the lower bound of the dimension. Our method can be used to deal with the sets with respect to more general targets and the sets based on the quantitative Poincaré recurrence properties.

Infinite iterated function systems with overlaps

2014 ◽  
Vol 36 (3) ◽  
pp. 890-907 ◽  
Author(s):  
SZE-MAN NGAI ◽  
JI-XI TONG
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function Systems ◽  
Topological Pressure ◽  
Separation Condition ◽  
Limit Sets ◽  
Weak Separation Condition ◽  
Iterated Function ◽  
Weak Separation ◽  
Separation Conditions

We formulate two natural but different extensions of the weak separation condition to infinite iterated function systems of conformal contractions with overlaps, and study the associated topological pressure functions. We obtain a formula for the Hausdorff dimension of the limit sets under these weak separation conditions.

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.

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