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논문 기본 정보
- 자료유형
- 학술저널
- 저자정보
- 저널정보
- 대한수학회 대한수학회지 대한수학회지 제59권 제1호
- 발행연도
- 2022.1
- 수록면
- 105 - 127 (23page)
- DOI
- 10.4134/JKMS.j210201
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초록· 키워드
오류제보하기
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.
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