Sensitivity of dendrite maps

Suppose that X   is a dendrite and f:X→Xf:X→X is a sensitive continuous map. We show that (a) (X,f)(X,f) contains a bilaterally transitive subsystem with nonempty interior; (b) the system (X,f)(X,f) satisfies only one of the following two conditions: (b1) (X,f)(X,f) contains a topologically transitive subsystem with nonempty interior; (b2) there exists an f-invariant nowhere dense closed subset A of X   such that the attraction basin Basin(A,f)Basin(A,f) contains a residual subset B   of an open set and the strong attraction basin Sbasin(A,f)Sbasin(A,f) is dense in B; (c) if X   is completely regular, then (X,f)(X,f) contains a relatively strongly mixing subsystem with nonempty interior, dense periodic points and positive topological entropy. Unlike for interval maps, we construct a sensitive dendrite map with zero topological entropy.