Generalized recurrence, compactifications, and the Lyapunov topology
We study generalized recurrence for closed relations on locally compact spaces. This includes continuous maps and real flows. The main tools are Lyapunov functions and their compactifications. Under certain conditions it is shown that the Lyapunov functions determine the topology of the space. In [2] a theory of generalized recurrence in dynamical systems is developed by means of continuous real-valued functions on the phase space. Let {φt} be a real action on the locally compact metric space X. Let L be a real-valued function on X which is non-decreasing on every orbit: L(φt(x)) ≥ L(x) for x ∈ X and t > 0. The generalized recurrent set R is defined to be those x ∈ X for which all such “Lyapunov” functions L are constant on the orbit of x. Clearly R contains the non-wandering points (and therefore the ordinary recurrent points). An intrinsic characterization of R can be obtained by means of prolongational limit sets. An interesting class of flows are those for which the generalized recurrent set is empty. Such flows are characterized by the existence of a Lyapunov function which is strictly increasing on every orbit: L(φt(x)) > L(x) for all x ∈ X and t > 0. It turns out that in this case the Lyapunov functions separate points, and so determine a Hausdorff topology on X. A natural question is whether this “Lyapunov” topology coincides with the original topology of X. Equivalently, if {xn} is a sequence in X, x ∈ X and L(xn)→ L(x) for all Lyapunov functions L, is it the case that xn → x? The purpose of this note is to answer this question in the affirmative (and in fact we will obtain a somewhat more general result). This will require an extensive detour into closed relations on locally compact metric spaces, as well as compactifications of such. In fact, we will obtain the corresponding result for closed relations (and as a consequence for continuous maps). 2010 Mathematics Subject Classification: Primary 54H20.