Sensitive Open MAP Semigroups On Peano Continua Having A Free Arc

Let X be a Peano continuum having a free arc and let C~0(X)be the semigroup of continuous self-maps of X.A subsemigroup F?C~0(X)is said to be sensitive,if there is some constant c>0 such that for any nonempty open set U?X,there is some f?F such that the diameter diam(f(U))>c.We show that if X admits a sensitive commutative subsemigroup F of C~0(X)consisting of continuous open maps,then either X is an arc,or X is a circle.