Sensitive semigroups of mappings on Peano continua having a free arc

Let X   be a Peano continuum having a free arc and let C0(X)C0(X) be the semigroup of continuous self-maps of X. It is shown that if there is a sensitive commutative subsemigroup F   of C0(X)C0(X) which consists of local homeomorphisms, then X is a circle.