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

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.