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

Department of Mathematics and Statistics, University of Prince Edward Island, 550 University Ave, Charlottetown, PE, C1A 4P3, Canada

E-mail Address: sislam@upei.ca

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STEPHEN CHANDLER

Department of Mathematics and Statistics, University of Prince Edward Island, 550 University Ave, Charlottetown, PE, C1A 4P3, Canada

E-mail Address: slwchandler@gmail.com

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https://doi.org/10.1142/S0218348X15500383Cited by:0 (Source: Crossref)

Abstract

Let $S$ 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

T = {τ 1 (x), τ 2 (x), \dots, τ K (x); p 1 (x), p 2 (x), \dots, p K (x)}, <math display="block" altimg="eq-00002.gif"><mrow><mi>T</mi><mo class="MathClass-rel">=</mo><mo class="MathClass-open" stretchy="false">{</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo><mo class="MathClass-op">\dots</mo><mo class="MathClass-punc">,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>K</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">;</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo><mo class="MathClass-op">\dots</mo><mo class="MathClass-punc">,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>K</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-close" stretchy="false">}</mo><mo class="MathClass-punc">,</mo></mrow></math>

where the probabilities

$p_{1} (x), p_{2} (x), \dots, p_{K} (x)$ of switching from one transformation to another are functions of positions, that is, at each step, the random map

$T$ moves the point

$x$ to

$τ_{k} (x)$ with probability

$p_{k} (x)$ . If the random map

$T$ has a unique invariant measure

$μ$ , then the support of

$μ$ is the attractor

$S$ . For a bounded region

$X \subseteq ℝ^{N}$ , we prove the existence of a sequence

${T_{0, n}^{*}}$ of IFSs with place-dependent probabilities whose invariant measures

${μ_{n}}$ are absolutely continuous with respect to Lebesgue measure. Moreover, if

$X$ is a compact metric space, we prove that

$μ_{n}$ converges weakly to

$μ$ as

$n \to \infty .$ We present examples with computations.

Keywords:

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