J. Korean Math. Soc. 2022; 59(1): 105-127
Online first article January 1, 2022 Printed January 1, 2022
https://doi.org/10.4134/JKMS.j210201
Copyright © The Korean Mathematical Society.
Byungik Kahng
7400 University Hills Boulevard
It is known that the maximal invariant set of a continuous iterative dynamical system in a compact Hausdorff space is equal to the intersection of its forward image sets, which we will call the it first minimal image set. In this article, we investigate the corresponding relation for a class of discontinuous self maps that are on the verge of continuity, or topologically almost continuous endomorphisms. We prove that the iterative dynamics of a topologically almost continuous endomorphisms yields a chain of minimal image sets that attains a unique transfinite length, which we call the maximal invariance order, as it stabilizes itself at the maximal invariant set. We prove the converse, too. Given ordinal number $\xi$, there exists a topologically almost continuous endomorphism $f$ on a compact Hausdorff space $X$ with the maximal invariance order $\xi$. We also discuss some further results regarding the maximal invariance order as more layers of topological restrictions are added.
Keywords: Maximal invariant set, discontinuous dynamics, ordinal numbers, transfinite induction and recursion
MSC numbers: Primary 37B99, 37E99, 03E10; Secondary 93C10, 93C65, 93B05
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