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

A homeomorphism S : X --> X of a compactum X with metric d is expansive if there is c > 0 such that if x,y is an element of X and x not equal y, then there is an integer n is an element of Z such that d(f(n)(x), f(n)(y)) > c. It is well-known that p- adic solenoids S-p (p greater than or equal to 2) admit expansive homeomorphisms, each S-p is an indecomposable continuum, and S-p cannot be embedded into the plane. In case of plane continua, the following interesting problem remains open: For each 1 less than or equal to n less than or equal to 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 S : X --> X is an expansive homeomorphism of a circle-like continuum X, then f is itself weakly chaotic in the sense of Devaney.