A Note on a Classical Ergodic Theorem for IFS

YANG Rongling; MA Dongkui

doi:10.6054/j.jscnun.2022064

Journal of South China Normal University (Natural Science Edition) > 2022 > 54(4): 109-112. > DOI: 10.6054/j.jscnun.2022064

YANG Rongling, MA Dongkui. A Note on a Classical Ergodic Theorem for IFS[J]. Journal of South China Normal University (Natural Science Edition), 2022, 54(4): 109-112. DOI: 10.6054/j.jscnun.2022064

Citation:

YANG Rongling, MA Dongkui. A Note on a Classical Ergodic Theorem for IFS[J]. Journal of South China Normal University (Natural Science Edition), 2022, 54(4): 109-112. DOI: 10.6054/j.jscnun.2022064

Citation:

YANG Rongling, MA Dongkui. A Note on a Classical Ergodic Theorem for IFS[J]. Journal of South China Normal University (Natural Science Edition), 2022, 54(4): 109-112. DOI: 10.6054/j.jscnun.2022064

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A Note on a Classical Ergodic Theorem for IFS

YANG Rongling¹,
MA Dongkui^2, ,

1.
School of Computer Engineering, Guangzhou City University of Technology, Guangzhou 510800, China
2.
School of Mathematics, South China University of Technology, Guangzhou 510640, China

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Received Date: January 09, 2022
Available Online: September 21, 2022

Graphical Abstract

Abstract

Abstract

The properties of the set of points where Elton's theorem does not hold are given. By constructing cylinder sets in symbolic space, it is proved that for a contractive IFS with probabilities and a continuous function g sa-tisfying that $\int_X g \mathrm{~d} \nu_1 \neq \int_X g \mathrm{~d} \nu_2$ , where ν₁ and ν₂ are any two different invariant measures of the IFS, the zero measure invariant set composed of points where Elton's theorem does not hold is either empty or carries full Hausdorff dimension and topological entropy.
- IFS,
- Elton's theorem,
- topological entropy,
- Hausdorff dimension

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