Stable Connectivity of Identity-Isotopic Gradient-Like Diffeomorphisms of Hyperbolic Surfaces

We consider a class of gradient-like diffeomorphisms of closed surfaces with negative Euler characteristic. It is shown that all such identity-isotopic diffeomorphisms are connected by a stable arc (containing finitely many saddle-node bifurcations). This result is contrast with the stable classification of gradient-like diffeomorphisms of the 2-sphere or the 2-torus, according to which the set of identity-isotopic diffeomorphisms on such surfaces is divided into a countable number of stable connectivity classes.