scholarly journals The absolute continuity of the invariant measure of random iterated function systems with overlaps

2010 ◽  
Vol 210 (1) ◽  
pp. 47-62
Author(s):  
Balázs Bárány ◽  
Tomas Persson
Keyword(s):  
Invariant Measure ◽  
Absolute Continuity ◽  
Iterated Function Systems ◽  
Iterated Function ◽  
The Absolute

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 ON INTERSECTING IFS FRACTALS WITH LINES

Fractals ◽  
2014 ◽  
Vol 22 (04) ◽  
pp. 1450014 ◽  
Author(s):  
JÓZSEF VASS
Keyword(s):  
Computer Graphics ◽  
Invariant Measure ◽  
Convex Hull ◽  
Infinite Number ◽  
Broad Class ◽  
Iterated Function Systems ◽  
Efficient Treatment ◽  
Intersection Problem ◽  
Iterated Function ◽  
Line Passing

IFS fractals — the attractors of Iterated Function Systems — have motivated plenty of research to date, partly due to their simplicity and applicability in various fields, such as the modeling of plants in computer graphics, and the design of fractal antennas. The statement and resolution of the Fractal-Line Intersection Problem is imperative for a more efficient treatment of certain applications. This paper intends to take further steps towards this resolution, building on the literature. For the broad class of hyperdense fractals, a verifiable condition guaranteeing intersection with any line passing through the convex hull of a planar IFS fractal is shown, in general ℝd for hyperplanes. The condition also implies a constructive algorithm for finding the points of intersection. Under certain conditions, an infinite number of approximate intersections are guaranteed, if there is at least one. Quantification of the intersection is done via an explicit formula for the invariant measure of IFS.

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.

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.

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