A Weak Equivalence and Topological Entropy

Abstract

In this paper we will investigate the topological entropies of mutually weakly equivalent topological flows. Roughly speaking, any two flows which are weakly equivalent to each other have the same orbits. So the notion of weak equivalence of flows is, in a sense, a generalization of time changes of flows. In [5] Totold investigated time changes of flows from a measure theoretical point of view. Especially he showed that for metrical (=measure theoretical) flows time changes preserve the properties that the metrical entropy is zero, positive, finite or infinite respectively. Here we will be rather concerned with topological flows and their topological entropies. First we will consider flows without fixed point. In this case we obtain a result analogous to Totoki's one. Namely, the properties that the topological entropy is zero, positive, finite or infinite respectively are invariant under weak equivalence of flows without fixed point (Theorem 1 in §3). But this is not the case if the flows have fixed points. Indeed, we will construct a pair of flows with the same orbits and a fixed point one of which has a positive entropy and the other has zero entropy (Theorem 2 in §4). In the proof of these two theorems we will appeal to a measure theoretical method. The point is that the topological entropy is the supremum of metrical entopies with respect to all invariant Borel probability measures (Lemma 2 in §3). The idea of the construction of the example in Section 4 to prove Theorem 2 is as follows. Take a flow with a fixed point such that each orbit visits a neighbourhood of the fixed point infinitely often and that the ratio of the sojourn time in the neighbourhood is uniformly positive. One can construct such a flow with a positive topological entropy. Then, lowering the speed of the flow