The nonexistence of expansive homeomorphisms of a class of continua which contains all decomposable circle-like continua

Abstract

A homeomorphism f: X -* X of a compactum X with metric d is expansive if there is c > 0 such that if x, y E X and x 7& y, then there is an integer n E Z such that d(f n(x), fn (y)) > c. It is well-known that padic solenoids Sp (p > 2) admit expansive homeomorphisms, each Sp is an indecomposable continuum, and Sp cannot be embedded into the plane. In case of plane continua, the following interesting problem remains open: For each 1 < n < 3, does there exist a plane continuum X so that X admits an expansive homeomorphism and X separates the plane into n components? For the case n = 2, the typical plane continua are circle-like continua, and every decomposable circle-like continuum can be embedded into the plane. Naturally, one may ask the following question: Does there exist a decomposable circle-like continuum admitting expansive homeomorphisms? In this paper, we prove that a class of continua, which contains all chainable continua, some continuous curves of pseudo-arcs constructed by W. Lewis and all decomposable circle-like continua, admits no expansive homeomorphisms. In particular, any decomposable circle-like continuum admits no expansive homeomorphism. Also, we show that if f: X -* X is an expansive homeomorphism of a circlelike continuum X, then f is itself weakly chaotic in the sense of Devaney.