Generic uniformly continuous mappings on unbounded hyperbolic spaces

We consider a complete, unbounded hyperbolic metric space X and a concave, nonzero and nondecreasing function ω:[0,+∞)→[0,+∞) with ω(0)=0 and study the space Cω(X) of uniformly continous self-mappings on X whose modulus of continuity is bounded from above by ω. We endow Cω(X) with the topology of uniform convergence on bounded sets and prove that the modulus of continuity of a generic mapping in Cω(X), in the sense of Baire categories, is precisely ω. Some related results in spaces of bounded mappings and in the topology of pointwise convergence are also discussed. This note can be seen as a completion of various results due to F. Strobin, S. Reich, A.J. Zaslavski, C. Bargetz and D. Thimm.