Sensitivity Of Dendrite Maps

Let X be a dendrite and let f:Xâ†'X be a sensitive continuous map. We show that (a) (X,f) must contain a bilaterally transitive subsystem with nonempty interior; (b) (X,f) either contains a topologically transitive subsystem with nonempty interior, or contains an f-invariant nowhere dense closed set A such that the attraction basin Basin(A,f) contains a residual set B with the strong attraction basin Sbasin(A,f) being dense in B; (c) If the closure of the branch point set of X contains no nondegenerate arc, then (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.