On a question of A. Kameyama concerning self-similar metrics

A. Kameyama introduced the concept of self-similar topological system and asked the following fundamental question: given a topological self-similar system (K,(fi)i∈{1,2,...,N}), does there exist a metric on K comparable to the topology such that all the functions fi are contractions? We modify Kameyama's question (which has a negative answer for an arbitrary topological self-similar system) by weakening the requirement that the functions in the topological self-similar system be contractions to requiring that they be φ-contractions. More precisely we give an affirmative answer to the following question: given a topological self-similar system (K,(fi)i∈{1,2,...,N}) does there exist a metric δ on K which is compatible with the original topology and a comparison function φ:[0,∞)→[0,∞) such that all the functions fi:(K,δ)→(K,δ) are φ-contractions? Consequently the iterated function system ((K,d),(fi)i∈{1,2,...,N}), where d is a metric on K compatible with the original topology on K, is φ-hyperbolic.