scholarly journals The Projection Pressure for Asymptotically Weak Separation Condition and Bowen's Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Chenwei Wang ◽  
Ercai Chen
Keyword(s):  
Variational Principle ◽  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Canonical Projection ◽  
Separation Condition ◽  
Weak Separation Condition ◽  
Iterated Function ◽  
Conformal Iterated Function System ◽  
Weak Separation ◽  
Projection Pressure

Let{Si}i=1lbe a weakly conformal iterated function system onRdwith attractorK. Letπbe the canonical projection. In this paper we define a new concept called “projection pressure”Pπ(φ)forφ∈C(Σ)and show the variational principle about the projection pressure under AWSC. Furthermore, we check that the zero of “projection pressure” still satisfies Bowen's equation. Using the root of Bowen's equation, we can get the Hausdorff dimension of the attractorK.

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.

The intersections of self-similar and self-affine sets with their perturbations under the weak separation condition

2016 ◽  
Vol 38 (4) ◽  
pp. 1353-1368 ◽  
Author(s):  
QI-RONG DENG ◽  
XIANG-YANG WANG
Keyword(s):  
Iterated Function System ◽  
The Self ◽  
Function System ◽  
Separation Condition ◽  
Weak Separation Condition ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Affine Set ◽  
Totally Disconnected

For a self-similar or self-affine iterated function system (IFS), let$\unicode[STIX]{x1D707}$be the self-similar or self-affine measure and$K$be the self-similar or self-affine set. Assume that the IFS satisfies the weak separation condition and$K$is totally disconnected; then, by using the technique of neighborhood decomposition, we prove that there is a neighborhood$\unicode[STIX]{x1D6FA}$of the identity map Id such that$\sup \{\unicode[STIX]{x1D707}(g(K)\cap K):g\in \unicode[STIX]{x1D6FA}\setminus \{\text{Id}\}\}<1$.

THE AVERAGE DENSITY OF SELF-CONFORMAL MEASURES

2001 ◽  
Vol 63 (3) ◽  
pp. 721-734 ◽  
Author(s):  
M. ZÄHLE
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Iterated Function System ◽  
Main Tool ◽  
Average Density ◽  
Energy Integral ◽  
Almost Everywhere ◽  
Iterated Function ◽  
Conformal Measures ◽  
Conformal Iterated Function System

The paper calculates the average density of the normalized Hausdorff measure on the fractal set generated by a conformal iterated function system. It equals almost everywhere a positive constant given by a truncated generalized s-energy integral, where s is the corresponding Hausdorff dimension. As a main tool a conditional Gibbs measure is determined. The appendix proves an appropriate extension of Birkhoff's ergodic theorem which is also of independent interest.

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 Dimension approximation of attractors of graph directed IFSs by self-similar sets

2018 ◽  
Vol 167 (01) ◽  
pp. 193-207 ◽  
Author(s):  
ÁBEL FARKAS
Keyword(s):  
Iterated Function System ◽  
Black Box ◽  
Function System ◽  
Separation Condition ◽  
Iterated Function ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Distance Sets

AbstractWe show that for the attractor (K1, . . ., Kq) of a graph directed iterated function system, for each 1 ⩽ j ⩽ q and ϵ &gt; 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dimHKj − ϵ &lt; dimHK. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property as a ‘black box’ we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets.

scholarly journals Hölder differentiability of self-conformal devil's staircases

2014 ◽  
Vol 156 (2) ◽  
pp. 295-311 ◽  
Author(s):  
SASCHA TROSCHEIT
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Hausdorff Dimension ◽  
Probability Distribution Function ◽  
Iterated Function System ◽  
Gibbs Measures ◽  
General Sense ◽  
Multifractal Formalism ◽  
Iterated Function ◽  
Set Of Points

AbstractIn this paper we consider the probability distribution function of a Gibbs measure supported on a self-conformal set given by an iterated function system (devil's staircase) applied to a compact subset of ${\mathbb R}$. We use thermodynamic multifractal formalism to calculate the Hausdorff dimension of the sets Sα0, Sα∞ and Sα, the set of points at which this function has, respectively, Hölder derivative 0, ∞ or no derivative in the general sense. This extends recent work by Darst, Dekking, Falconer, Kesseböhmer and Stratmann, and Yao, Zhang and Li by considering arbitrary such Gibbs measures given by a potential function independent of the geometric potential.

scholarly journals Resonance between Cantor sets

2009 ◽  
Vol 29 (1) ◽  
pp. 201-221 ◽  
Author(s):  
YUVAL PERES ◽  
PABLO SHMERKIN
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Unit Interval ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Scaling Parameter ◽  
Iterated Function ◽  
Self Similar ◽  
Image Position ◽  
Geometric Resonance

AbstractLet Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two intervals. We prove that if log b/log a is irrational, then where dim is Hausdorff dimension. More generally, given two self-similar sets K,K′ in ℝ and a scaling parameter s>0, if the dimension of the arithmetic sum K+sK′ is strictly smaller than dim (K)+dim (K′)≤1 (‘geometric resonance’), then there exists r<1 such that all contraction ratios of the similitudes defining K and K′ are powers of r (‘algebraic resonance’). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

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.

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