Maximal invariance of topologically almost continuous iterative dynamics
J. Korean Math. Soc. 2022 Vol. 59, No. 1, 105-127
Published online November 15, 2021
Printed January 1, 2022
Byungik Kahng
7400 University Hills Boulevard
Abstract : 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 {\it 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 {\it length}, which we call the {\it 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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