Ergodic properties of random iterations of analytic functions

1999 ◽  
Vol 19 (6) ◽  
pp. 1379-1388
Author(s):  
AMIRAN AMBROLADZE
Keyword(s):  
Complex Plane ◽  
Analytic Functions ◽  
Entire Functions ◽  
Iterated Function System ◽  
Ergodic Property ◽  
Function System ◽  
Ergodic Properties ◽  
Iterated Function ◽  
Extended Complex ◽  
Extended Complex Plane

It is a known fact that an iterated function system (IFS) of entire functions is not necessarily ergodic. In this paper we show that if an IFS of analytic functions is defined in a domain whose boundary contains more than two points (in the extended complex plane) then the system possesses an ergodic property.

A classical ergodic property for IFS: a simple proof

1998 ◽  
Vol 18 (3) ◽  
pp. 609-611 ◽  
Author(s):  
B. FORTE ◽  
F. MENDIVIL
Keyword(s):  
Invariant Measure ◽  
Simple Proof ◽  
Iterated Function System ◽  
Ergodic Property ◽  
Function System ◽  
Iterated Function ◽  
Considerable Simplification ◽  
Contraction Maps

Let $\{w_i,p_i\}$ be a contractive iterated function system (IFS) [1, pp. 79–80] with probabilities, i.e. a set of contraction maps $w_i:X\to X$ with associated probabilities $p_i$, $i=1,2,\ldots,N$. We provide a simple proof that for almost every address sequence $\sigma$ and for all $x$ the limit $\lim_{n\to \infty}1/n\sum_{i\le n}f(w_{\sigma_n}\circ w_{\sigma_{n-1}}\circ\cdots\circ w_{\sigma_1}(x))$ exists and is equal to $\int_Xf(z)\,d\mu(z)$, where $\mu$ is the invariant measure of the IFS. This is the so called ‘ergodic property’ for the IFS and was proved by Elton in [3]. However, the uniqueness of the invariant measure was not previously exploited. This provides considerable simplification to the proof.

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].

Stochastic Fractal Modeling and Process Planning for Machining Using Two-Dimensional Iterated Function System

2008 ◽  
Vol 392-394 ◽  
pp. 575-579
Author(s):  
Yu Hao Li ◽  
Jing Chun Feng ◽  
Y. Li ◽  
Yu Han Wang
Keyword(s):  
Path Planning ◽  
Tool Path ◽  
Iterated Function System ◽  
Numerical Control ◽  
Function System ◽  
Tool Path Planning ◽  
Ongoing Research ◽  
Iterated Function ◽  
Planning Algorithm ◽  
Path Planning Algorithm

Self-affine and stochastic affine transforms of R2 Iterated Function System (IFS) are investigated in this paper for manufacturing non-continuous objects in nature that exhibit fractal nature. A method for modeling and fabricating fractal bio-shapes using machining is presented. Tool path planning algorithm for numerical control machining is presented for the geometries generated by our fractal generation function. The tool path planning algorithm is implemented on a CNC machine, through executing limited number of iteration. This paper describes part of our ongoing research that attempts to break through the limitation of current CAD/CAM and CNC systems that are oriented to Euclidean geometry objects.

Fractal dimension and iterated function system (IFS) for speech recognition

1992 ◽  
Vol 28 (15) ◽  
pp. 1382 ◽  
Author(s):  
E.L.J. Bohez ◽  
T.R. Senevirathne ◽  
J.A. van Winden
Keyword(s):  
Fractal Dimension ◽  
Speech Recognition ◽  
Iterated Function System ◽  
Function System ◽  
Iterated Function

Fractal Top for Contractive and Non-Contractive Transformations

2012 ◽  
Vol 3 (4) ◽  
pp. 49-65
Author(s):  
Sarika Jain ◽  
S. L. Singh ◽  
S. N. Mishra
Keyword(s):  
Fixed Point ◽  
Iterated Function System ◽  
Function System ◽  
Feedback Process ◽  
Iterated Function ◽  
Spatial Part ◽  
One Step ◽  
The One ◽  
Definition Of

Barnsley (2006) introduced the notion of a fractal top, which is an addressing function for the set attractor of an Iterated Function System (IFS). A fractal top is analogous to a set attractor as it is the fixed point of a contractive transformation. However, the definition of IFS is extended so that it works on the colour component as well as the spatial part of a picture. They can be used to colour-render pictures produced by fractal top and stealing colours from a natural picture. Barnsley has used the one-step feed- back process to compute the fractal top. In this paper, the authors introduce a two-step feedback process to compute fractal top for contractive and non-contractive transformations.

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