Bounded recurrent sets for planar homeomorphisms. (English) Zbl 1031.37038

Le Roux, Frédéric

Ergodic Theory Dyn. Syst. 19, No. 4, 1085-1091 (1999).

Summary: We answer a question proposed by M. M. Barge and J. M. Franks [in: M. W. Hirsch (ed.) et al., From topology to computation: Proceedings of the Smalefest (Berkeley, CA, 1990), New York: Springer-Verlag, 186-195 (1993; Zbl 0806.57005)] by showing that a bounded connected set in the plane which is closed or open and meets an infinite number of its iterates under an orientation-preserving fixed point free homeomorphism must meet all of them.

MSC:

37E30	Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37B20	Notions of recurrence and recurrent behavior in topological dynamical systems
54H10	Topological representations of algebraic systems

Keywords:

orientation-preserving homeomorphism; bounded connected set in the plane

Citations:

Zbl 0806.57005