2024/04/16 by Mai, Jiehua, Yan, Kesong, Zeng, Fanping
#37B20 #37C25 #37E30 #54H20 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2404.10247
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 \citeFa, 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.