Orientation Preserving Homeomorphisms of the Plane having BP-Chain Recurrent Points
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| Format: | Preprint |
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2024
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| _version_ | 1866914756564090880 |
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| author | Mai, Jiehua Yan, Kesong Zeng, Fanping |
| author_facet | Mai, Jiehua Yan, Kesong Zeng, Fanping |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_10247 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orientation Preserving Homeomorphisms of the Plane having BP-Chain Recurrent Points Mai, Jiehua Yan, Kesong Zeng, Fanping Dynamical Systems 37E30, 37C25, 37B20, 54H20 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. |
| title | Orientation Preserving Homeomorphisms of the Plane having BP-Chain Recurrent Points |
| topic | Dynamical Systems 37E30, 37C25, 37B20, 54H20 |
| url | https://arxiv.org/abs/2404.10247 |