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 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.

scholarly journals Graph directed Markov systems on Hilbert spaces

2009 ◽  
Vol 147 (2) ◽  
pp. 455-488 ◽  
Author(s):  
R. D. MAULDIN ◽  
T. SZAREK ◽  
M. URBAŃSKI
Keyword(s):  
Hausdorff Measure ◽  
Iterated Function System ◽  
Sufficient Conditions ◽  
Topological Pressure ◽  
Limit Set ◽  
Covering Theorem ◽  
Limit Sets ◽  
Markov Systems ◽  
Polish Spaces ◽  
Iterated Function

AbstractWe deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowen's formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and provide sufficient conditions for geometric perfectness. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be positive. As a by–product of the mainstream of our investigations we prove a 4r–covering theorem for all metric spaces. It enables us to establish appropriate co–Frostman type theorems.

Local dimension for piecewise monotonic maps on the interval

1995 ◽  
Vol 15 (6) ◽  
pp. 1119-1142 ◽  
Author(s):  
Franz Hofbauer
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measures ◽  
Positive Characteristic ◽  
Characteristic Exponent ◽  
Local Dimension ◽  
Pressure Function ◽  
Piecewise Monotonic ◽  
Almost Everywhere ◽  
Conformal Measures

AbstractThe local dimension of invariant and conformal measures for piecewise monotonic transformations on the interval is considered. For ergodic invariant measures m with positive characteristic exponent χm we show that the local dimension exists almost everywhere and equals hm/χm For certain conformal measures we show a relation between a pressure function and the Hausdorff dimension of sets, on which the local dimension is constant.

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.

BOUNDS FOR DIMENSION OF THE ATTRACTOR OF A SCALED IFS

2013 ◽  
Vol 06 (02) ◽  
pp. 1350028 ◽  
Author(s):  
S. Minirani ◽  
Sunil Mathew
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Complete Metric Space ◽  
Upper Bounds ◽  
Scaling Factor ◽  
Lower And Upper Bounds ◽  
Iterated Function ◽  
Finite Set ◽  
Existence Conditions ◽  
Similarity Dimension

A hyperbolic iterated function system (IFS) consists of a complete metric space X together with a finite set of contraction mappings on X. In this paper, the notion of scaled IFS is defined and its existence conditions are examined. The relation between the similarity dimension of the attractors of a given homogeneous IFS and a scaled IFS and its dependency on the scaling factor are studied. A lower and upper bounds for the Hausdorff dimension of the attractor of a scaled IFS is obtained.

scholarly journals VISIBILITY OF CARTESIAN PRODUCTS OF CANTOR SETS

Fractals ◽  
2020 ◽  
Vol 28 (06) ◽  
pp. 2050119
Author(s):  
TINGYU ZHANG ◽  
KAN JIANG ◽  
WENXIA LI
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Cantor Sets ◽  
Function System ◽  
Topological Properties ◽  
Cartesian Products ◽  
Iterated Function

Let [Formula: see text] be the attractor of the following iterated function system(IFS): [Formula: see text] Given [Formula: see text], we say the line [Formula: see text] is visible through [Formula: see text] if [Formula: see text] Let [Formula: see text]. In this paper, we give a complete description of [Formula: see text], containing its Hausdorff dimension and topological properties.

scholarly journals Dimensions of random affine code tree fractals

2013 ◽  
Vol 34 (3) ◽  
pp. 854-875 ◽  
Author(s):  
ESA JÄRVENPÄÄ ◽  
MAARIT JÄRVENPÄÄ ◽  
ANTTI KÄENMÄKI ◽  
HENNA KOIVUSALO ◽  
ÖRJAN STENFLO ◽  
...  
Keyword(s):  
Hausdorff Dimension ◽  
General Class ◽  
Classical Result ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Probability Measures ◽  
Box Counting ◽  
Iterated Function ◽  
Natural Measures

AbstractWe study the dimension of code tree fractals, a class of fractals generated by a set of iterated function systems. We first consider deterministic affine code tree fractals, extending to the code tree fractal setting the classical result of Falconer and Solomyak on the Hausdorff dimension of self-affine fractals generated by a single iterated function system. We then calculate the almost sure Hausdorff, packing and box counting dimensions of a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.

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