A Free Open Mapping Of The Peano Continuum With Free Arcs

Early in 1988, Kawamura showed that each Peano continuum with a free arc admits no expansive homeomorphisms in [1]. It is known that expansivity is stronger than sensitivity. A natural problem is whether such a continuum admits a sensitive homeomorphism. In [2], Mai and Shi gave a negative answer in the more general case. They proved that a Peano continuum having a free arc admits no sensitive commutative group actions.However, this kind of continuum may admit a sensitive open map. A simple example is the circle map f : S1â†'S1, x â†' 2x, for all x ∈S1= R/Z.In this paper, we mainly study the sensitive open maps on Peano continua having a free arc, and we prove the following theorem.The main theorem: Let X be a Peano continuum having a free arc. If X admits a sensitive open map, then either X is homeomorphic to the close interval [0, 1] or X is homeomorphic to the unit circle S1.In addition, we give an example of a semi-open map on a Peano continuum having a free arc, which is sensitive. This example shows that the theorem above cannot be generalized to semi-open maps.