Common fixed points of set-valued mappings.
In this paper we study the commutativity property for topological sequence entropy. We prove that if is a compact metric space and are continuous maps then for every increasing sequence if , and construct a counterexample for the general case. In the interim, we also show that the equality is true if but does not necessarily hold if is an arbitrary compact metric space.
A family f₁,..., fₙ of operators on a complete metric space X is called contractive if there exists a positive λ < 1 such that for any x,y in X we have for some i. Austin conjectured that any commuting contractive family of operators has a common fixed point, and he proved this for the case of two operators. We show that Austin’s conjecture is true for three operators, provided that λ is sufficiently small.