scholarly journals Dimensions of Fractals Generated by Bi-Lipschitz Maps

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qi-Rong Deng ◽  
Sze-Man Ngai
Keyword(s):  
Iterated Function Systems ◽  
Topological Pressure ◽  
Separation Condition ◽  
Lipschitz Mappings ◽  
Affine Maps ◽  
Lipschitz Maps ◽  
Iterated Function ◽  
Bounded Distortion ◽  
Box Dimensions

On the class of iterated function systems of bi-Lipschitz mappings that are contractions with respect to some metrics, we introduce a logarithmic distortion property, which is weaker than the well-known bounded distortion property. By assuming this property, we prove the equality of the Hausdorff and box dimensions of the attractor. We also obtain a formula for the dimension of the attractor in terms of certain modified topological pressure functions, without imposing any separation condition. As an application, we prove the equality of Hausdorff and box dimensions for certain iterated function systems consisting of affine maps and nonsmooth maps.

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.

Regularity of Hausdorff measure function for conformal dynamical systems

2016 ◽  
Vol 160 (3) ◽  
pp. 537-563 ◽  
Author(s):  
MARIUSZ URBAŃSKI ◽  
ANNA ZDUNIK
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Measure ◽  
Hölder Continuity ◽  
Iterated Function Systems ◽  
Separation Condition ◽  
Holder Continuity ◽  
Measure Function ◽  
Iterated Function ◽  
Strong Separation ◽  
Strong Separation Condition

AbstractWe deal with the question of continuity of numerical values of Hausdorff measures in parametrised families of linear (similarity) and conformal dynamical systems by developing the pioneering work of Lars Olsen and the work [SUZ]. We prove Hölder continuity of the function ascribing to a parameter the numerical value of the Hausdorff measure of either the corresponding limit set or the corresponding Julia set. We consider three cases. Firstly, we consider the case of parametrised families of conformal iterated function systems in $\mathbb{R}$k with k ⩾ 3. Secondly, we consider all linear iterated function systems consisting of similarities in $\mathbb{R}$k with k ⩾ 1. In either of these two cases, the strong separation condition is assumed. In the latter case the Hölder exponent obtained is equal to 1/2. Thirdly, we prove such Hölder continuity for analytic families of conformal expanding repellers in the complex plane $\mathbb{C}$. Furthermore, we prove the Hausdorff measure function to be piecewise real–analytic for families of naturally parametrised linear IFSs in $\mathbb{R}$ satisfying the strong separation condition. On the other hand, we also give an example of a family of linear IFSs in $\mathbb{R}$ for which this function is not even differentiable at some parameters.

scholarly journals Box dimension for graphs of fractal functions

1997 ◽  
Vol 40 (2) ◽  
pp. 331-344
Author(s):  
Gavin Brown ◽  
Qinghe Yin
Keyword(s):  
Iterated Function Systems ◽  
Box Dimension ◽  
Affine Maps ◽  
Iterated Function ◽  
Fractal Functions

We calculate the box-dimension for a class of nowhere differentiable curves defined by Markov attractors of certain iterated function systems of affine maps.

A new class of markov processes for image encoding

1988 ◽  
Vol 20 (01) ◽  
pp. 14-32 ◽  
Author(s):  
Michael F. Barnsley ◽  
John H. Elton
Keyword(s):  
Invariant Measures ◽  
Metric Spaces ◽  
Iterated Function Systems ◽  
Interesting Case ◽  
Image Encoding ◽  
New Class ◽  
Affine Maps ◽  
Iterated Maps ◽  
Iterated Function ◽  
Infinite Extent

A new class of iterated function systems is introduced, which allows for the computation of non-compactly supported invariant measures, which may represent, for example, greytone images of infinite extent. Conditions for the existence and attractiveness of invariant measures for this new class of randomly iterated maps, which are not necessarily contractions, in metric spaces such as , are established. Estimates for moments of these measures are obtained. Special conditions are given for existence of the invariant measure in the interesting case of affine maps on . For non-singular affine maps on , the support of the measure is shown to be an infinite interval, but Fourier transform analysis shows that the measure can be purely singular even though its distribution function is strictly increasing.

A new class of markov processes for image encoding

1988 ◽  
Vol 20 (1) ◽  
pp. 14-32 ◽  
Author(s):  
Michael F. Barnsley ◽  
John H. Elton
Keyword(s):  
Invariant Measures ◽  
Metric Spaces ◽  
Iterated Function Systems ◽  
Interesting Case ◽  
Image Encoding ◽  
New Class ◽  
Affine Maps ◽  
Iterated Maps ◽  
Iterated Function ◽  
Infinite Extent

A new class of iterated function systems is introduced, which allows for the computation of non-compactly supported invariant measures, which may represent, for example, greytone images of infinite extent. Conditions for the existence and attractiveness of invariant measures for this new class of randomly iterated maps, which are not necessarily contractions, in metric spaces such as , are established. Estimates for moments of these measures are obtained.Special conditions are given for existence of the invariant measure in the interesting case of affine maps on . For non-singular affine maps on , the support of the measure is shown to be an infinite interval, but Fourier transform analysis shows that the measure can be purely singular even though its distribution function is strictly increasing.

scholarly journals Piecewise linear iterated function systems on the line of overlapping construction

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 245-277
Author(s):  
R Dániel Prokaj ◽  
Károly Simon
Keyword(s):  
Hausdorff Dimension ◽  
Real Line ◽  
Piecewise Linear ◽  
Iterated Function Systems ◽  
Packing Dimension ◽  
Linear Functions ◽  
Separation Condition ◽  
Piecewise Linear Functions ◽  
Exceptional Set ◽  
Iterated Function

Abstract In this paper we consider iterated function systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS are injective, but we assume that their derivatives are separated from zero. We prove that if we fix all the slopes but perturb all other parameters, then for all parameters outside of an exceptional set of less than full packing dimension, the Hausdorff dimension of the attractor is equal to the exponent which comes from the most natural system of covers of the attractor.

SOLVING THE INVERSE PROBLEM FOR FUNCTION/IMAGE APPROXIMATION USING ITERATED FUNCTION SYSTEMS II: ALGORITHM AND COMPUTATIONS

Fractals ◽  
1994 ◽  
Vol 02 (03) ◽  
pp. 335-346 ◽  
Author(s):  
BRUNO FORTE ◽  
EDWARD R. VRSCAY
Keyword(s):  
Inverse Problem ◽  
Quadratic Programming ◽  
Iterated Function Systems ◽  
Grey Level ◽  
Image Approximation ◽  
Affine Maps ◽  
Iterated Function ◽  
Contractive Maps ◽  
Target Set ◽  
Infinite Set

In this paper, we provide an algorithm for the construction of IFSM approximations to a target set [Formula: see text], where X ⊂ RD and µ = m(D) (Lebesgue measure). The algorithm minimizes the squared "collage distance" [Formula: see text]. We work with an infinite set of fixed affine IFS maps wi: X → X satisfying a certain density and nonoverlapping condition. As such, only an optimization over the grey level maps ϕi: R+ → R+ is required. If affine maps are assumed, i.e. ϕi = αit + βi, then the algorithm becomes a quadratic programming (QP) problem in the αi and βi. We can also define a "local IFSM" (LIFSM) which considers the actions of contractive maps wi on subsets of X to produce smaller subsets. Again, affine ϕi maps are used, resulting in a QP problem. Some approximations of functions on [0,1] and images in [0, 1]2 are presented.

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.

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