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arXiv · 2404.10247

Orientation Preserving Homeomorphisms of the Plane having BP-Chain Recurrent Points

Jiehua Mai·Kesong Yan·Fanping Zeng
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Abstract

More than a century ago, L. E. J. Brouwer proved a famous theorem, which says that any orientation preserving homeomorphism of the plane having a periodic point must have a fixed point. In recent years, there are still some authors giving various proofs of this fixed point theorem. In \cite{Fa}, Fathi showed that the condition``having a periodic point'' in this theorem can be weakened to ``having a non-wandering point''. In this paper, we first give a new proof of Brouwer's theorem, which is relatively more simpler and the statement is more compact. Further, we propose a notion of BP-chain recurrent points, which is a generalization of the concept of non-wandering points, and we prove that if an orientation preserving homeomorphism of the plane has a BP-chain recurrent point, then it has a fixed point. This further weakens the condition in the Brouwer's fixed point theorem on plane.

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Jiehua Mai, Kesong Yan, Fanping Zeng. 2024-04-16. Orientation Preserving Homeomorphisms of the Plane having BP-Chain Recurrent Points. https://arxiv.org/abs/2404.10247

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