Generic uniformly continuous mappings on unbounded hyperbolic spaces

We consider a complete, unbounded, hyperbolic metric space $X$ and a concave, nonzero and nondecreasing function $\omega :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ with $\omega (0)=0$ and study the space ${\mathcal{C}}_{\omega }(X)$ of uniformly continous self-mappings on $X$ whose modulus of continuity is bounded above by $\omega$. We endow ${\mathcal{C}}_{\omega }(X)$ with the topology of uniform convergence on bounded sets and prove that the modulus of continuity of the generic mapping in ${\mathcal{C}}_{\omega }(X)$, in the sense of Baire categories, is precisely $\omega$. 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. Zaslavski, C. Bargetz and D. Thimm.