scholarly journals On Hausdorff dimension of invariant measures arising from non-contractive iterated function systems

2002 ◽  
Vol 181 (2) ◽  
pp. 223-237 ◽  
Author(s):  
Józef Myjak ◽  
Tomasz Szarek
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measures ◽  
Iterated Function Systems ◽  
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.

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.

scholarly journals Ledrappier–Young formulae for a family of nonlinear attractors

2021 ◽  
Author(s):  
Natalia Jurga ◽  
Lawrence D. Lee
Keyword(s):  
Invariant Measures ◽  
Gibbs Measures ◽  
Iterated Function Systems ◽  
Hölder Continuous ◽  
Natural Class ◽  
Iterated Function ◽  
Bernoulli Measures

AbstractWe study a natural class of invariant measures supported on the attractors of a family of nonlinear, non-conformal iterated function systems introduced by Falconer, Fraser and Lee. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show that these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier–Young formula.

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.

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