RIGOROUS NUMERICAL ESTIMATION OF LYAPUNOV EXPONENTS AND INVARIANT MEASURES OF ITERATED FUNCTION SYSTEMS AND RANDOM MATRIX PRODUCTS

2000 ◽  
Vol 10 (01) ◽  
pp. 103-122 ◽  
Author(s):  
GARY FROYLAND ◽  
KAZUYUKI AIHARA
Keyword(s):  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Random Matrix ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Matrix Product ◽  
True Value ◽  
Iterated Function ◽  
Low Dimensional ◽  
Unique Invariant Measure

We present a fast, simple matrix method of computing the unique invariant measure and associated Lyapunov exponents of a nonlinear iterated function system. Analytic bounds for the error in our approximate invariant measure (in terms of the Hutchinson metric) are provided, while convergence of the Lyapunov exponent estimates to the true value is assured. As a special case, we are able to rigorously estimate the Lyapunov exponents of an iid random matrix product. Computation of the Lyapunov exponents is carried out by evaluating an integral with respect to the unique invariant measure, rather than following a random orbit. For low-dimensional systems, our method is considerably quicker and more accurate than conventional methods of exponent computation. An application to Markov random matrix product is also described.

APPROXIMATION BY ABSOLUTELY CONTINUOUS INVARIANT MEASURES OF ITERATED FUNCTION SYSTEMS WITH PLACE-DEPENDENT PROBABILITIES

Fractals ◽  
2015 ◽  
Vol 23 (04) ◽  
pp. 1550038
Author(s):  
MD SHAFIQUL ISLAM ◽  
STEPHEN CHANDLER
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Bounded Region ◽  
Compact Metric Space ◽  
Absolutely Continuous ◽  
Iterated Function ◽  
Unique Invariant Measure ◽  
Absolutely Continuous Invariant

Let [Formula: see text] be the attractor (fractal) of a contractive iterated function system (IFS) with place-dependent probabilities. An IFS with place-dependent probabilities is a random map [Formula: see text] where the probabilities [Formula: see text] of switching from one transformation to another are functions of positions, that is, at each step, the random map [Formula: see text] moves the point [Formula: see text] to [Formula: see text] with probability [Formula: see text]. If the random map [Formula: see text] has a unique invariant measure [Formula: see text], then the support of [Formula: see text] is the attractor [Formula: see text]. For a bounded region [Formula: see text], we prove the existence of a sequence [Formula: see text] of IFSs with place-dependent probabilities whose invariant measures [Formula: see text] are absolutely continuous with respect to Lebesgue measure. Moreover, if [Formula: see text] is a compact metric space, we prove that [Formula: see text] converges weakly to [Formula: see text] as [Formula: see text] We present examples with computations.

scholarly journals Approximating integrals with respect to stationary probability measures of iterated function systems

2020 ◽  
pp. 1-17
Author(s):  
ITALO CIPRIANO ◽  
NATALIA JURGA
Keyword(s):  
Lyapunov Exponents ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Unit Interval ◽  
Function System ◽  
Probability Measures ◽  
Stationary Probability ◽  
Iterated Function ◽  
Wasserstein Distances

We study fast approximation of integrals with respect to stationary probability measures associated to iterated function systems on the unit interval. We provide an algorithm for approximating the integrals under certain conditions on the iterated function system and on the function that is being integrated. We apply this technique to estimate Hausdorff moments, Wasserstein distances and Lyapunov exponents of stationary probability measures.

Continuity of Attractors and Invariant Measures for Iterated Function Systems

1994 ◽  
Vol 37 (3) ◽  
pp. 315-329 ◽  
Author(s):  
P. M. Centore ◽  
E. R. Vrscay
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Iterated Function System ◽  
Julia Sets ◽  
Iterated Function Systems ◽  
Function System ◽  
Iterated Function ◽  
Quadratic Complex

AbstractWe prove the "folklore" results that both the attractor A and invariant measure μ of an N-map Iterated Function System (IFS) vary continuously with variations in the contractive IFS maps as well as the probabilities. This represents a generalization of Barnsley's result showing the continuity of attractors with respect to variations of a parameter appearing in the IFS maps. Some applications are presented, including approximations of attractors and invariant measures of nonlinear IFS, as well as some novel approximations of Julia sets for quadratic complex maps.

ARCLENGTH AS THE INVARIANT MEASURE FOR AN IFS WITH PROBABILITIES

Fractals ◽  
2015 ◽  
Vol 23 (04) ◽  
pp. 1550046
Author(s):  
D. LA TORRE ◽  
F. MENDIVIL
Keyword(s):  
Invariant Measure ◽  
Iterated Function System ◽  
Function System ◽  
Iterated Function

Given a continuous rectifiable function [Formula: see text], we present a simple Iterated Function System (IFS) with probabilities whose invariant measure is the normalized arclength measure on the graph of [Formula: see text].

scholarly journals Exponential Convergence For Markov Systems

2015 ◽  
Vol 29 (1) ◽  
pp. 139-149 ◽  
Author(s):  
Maciej Ślęczka
Keyword(s):  
Invariant Measure ◽  
Exponential Convergence ◽  
Iterated Function Systems ◽  
Markov Systems ◽  
Markov Operators ◽  
Iterated Function

AbstractMarkov operators arising from graph directed constructions of iterated function systems are considered. Exponential convergence to an invariant measure is proved.

scholarly journals Generalized Iterated Function Systems Containing Functions of Integral Type

2018 ◽  
Vol 7 (3.31) ◽  
pp. 126
Author(s):  
Minirani S ◽  
. .
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Product Space ◽  
Iterated Function System ◽  
Complete Metric Space ◽  
Iterated Function Systems ◽  
Function System ◽  
Finite Collection ◽  
Integral Type ◽  
Iterated Function

A finite collection of mappings which are contractions on a complete metric space constitutes an iterated function system. In this paper we study the generalized iterated function system which contain generalized contractions of integral type from the product space . We prove the existence and uniqueness of the fixed point of such an iterated function system which is also known as its attractor. 

Fractals in the Large

1998 ◽  
Vol 50 (3) ◽  
pp. 638-657 ◽  
Author(s):  
Robert S. Strichartz
Keyword(s):  
Invariant Measure ◽  
Power Series ◽  
Metric Space ◽  
Iterated Function System ◽  
Invariant Set ◽  
Invariant Sets ◽  
Iterated Function ◽  
Positive Weights ◽  
Self Similar ◽  
Expansive Maps

AbstractA reverse iterated function system (r.i.f.s.) is defined to be a set of expansive maps ﹛T1,…, Tm﹜ on a discrete metric space M. An invariant set F is defined to be a set satisfying , and an invariant measure μ is defined to be a solution of for positive weights pj. The structure and basic properties of such invariant sets and measures is described, and some examples are given. A blowup ℱ of a self-similar set F in ℝn is defined to be the union of an increasing sequence of sets, each similar to F. We give a general construction of blowups, and show that under certain hypotheses a blowup is the sum set of F with an invariant set for a r.i.f.s. Some examples of blowups of familiar fractals are described. If μ is an invariant measure on ℤ+ for a linear r.i.f.s., we describe the behavior of its analytic transform, the power series on the unit disc.

EXISTENCE AND BOX DIMENSION OF GENERAL RECURRENT FRACTAL INTERPOLATION FUNCTIONS

2020 ◽  
pp. 1-13
Author(s):  
HUO-JUN RUAN ◽  
JIAN-CI XIAO ◽  
BING YANG
Keyword(s):  
Iterated Function System ◽  
General Setting ◽  
Iterated Function Systems ◽  
Box Dimension ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Recurrent System ◽  
Definition Of ◽  
Interpolation Functions

Abstract The notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley et al. [‘Recurrent iterated function systems’, Constr. Approx.5 (1989), 362–378]. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on $\mathbb {R}^2$ . We generalise the definition of RFIFs so that iterated functions in the recurrent system need not be contractive with respect to the first variable. We obtain the box dimensions of all self-affine RFIFs in this general setting.

Accessibility on iterated function systems

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Maliheh Mohtashamipour ◽  
Alireza Zamani Bahabadi
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Sufficient Condition ◽  
Skew Product ◽  
Iterated Function ◽  
Topologically Mixing ◽  
Mixing Property

AbstractIn this paper, we define accessibility on an iterated function system (IFS) and show that it provides a sufficient condition for the transitivity of this system and its corresponding skew product. Then, by means of a certain tool, we obtain the topologically mixing property. We also give some results about the ergodicity and stability of accessibility and, further, illustrate accessibility by some examples.

Some Results on Recurrent Fractal Interpolation Function

2020 ◽  
Vol 12 (8) ◽  
pp. 1038-1043
Author(s):  
Wadia Faid Hassan Al-Shameri
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Data Set ◽  
Fractal Interpolation Function ◽  
Iterated Function ◽  
Fractal Functions ◽  
Fractal Approximation

Barnsley (Barnsley, M.F., 1986. Fractal functions and interpolation. Constr. Approx., 2, pp.303–329) introduced fractal interpolation function (FIF) whose graph is the attractor of an iterated function system (IFS) for describing the data that have an irregular or self-similar structure. Barnsley et al. (Barnsley, M.F., et al., 1989. Recurrent iterated function systems in fractal approximation. Constr. Approx., 5, pp.3–31) generalized FIF in the form of recurrent fractal interpolation function (RFIF) whose graph is the attractor of a recurrent iterated function system (RIFS) to fit data set which is piece-wise self-affine. The primary aim of the present research is investigating the RFIF approach and using it for fitting the piece-wise self-affine data set in ℜ2.

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