Metrization and Liapunov functions. V (Q1111194)

This paper is the continuation of the papers reviewed in Zbl 0548.54028; Zbl 0562.54060 and Zbl 0658.54026. The problem we deal with is the possibility of constructing remetrizations of the phase space such that the new metrics describe attraction and repulsion properties of the trajectories. An obstacle to this approach is the presence of nonequilibrium recurrent trajectories (precluding the existence of global Liapunov functions). This difficulty can be overcome by factorization according to Auslander recurrence classes. Now we concern ourselves with an additional obstacle. Replacing the finiteness condition of a previous result by local finitness, we prove a theorem on the existence of metrics of Liapunov type. On the other hand, we point out by examples that, in general, local finiteness can not be weakened further. The dynamical system defined in Example 1 has no recurrent trajectories but equilibrium points. Example 2 shows that the local finiteness condition can not be dropped even if compact isolated invariant sets are replaced by asymptotically stable equilibrium points.