Trends in Mathematics - New Series Information Center for Mathematical Sciences Volume 10, Number 2, 2008, pages 121–130 2008 International Workshop on Dynamical Systems and Related Topics c 2008 ICMS in KAIST ENTROPY AND MEASURE DEGENERACY FOR FLOWS WENXIANG SUN* AND EDSON VARGAS** Abstract. In discrete dynamical systems topological entropy is a topological invariant and a measurement of the complexity of a system. In continuous dynamical systems, in general, topological entropy defined as usual by the time one map does not work so well in what concerns these aspects. The point is that the natural notion of equivalence in the discrete case is topological conjugacy which preserves time while in the continuous case the natural notion of equivalence is topological equivalence which allow reparametrizations of the orbits. The main issue happens in the case that the system has fixed points and will be our subject here. 1. Introduction Two continuous flows defined on a compact metric space are topologically equiva- lent if there exists a homeomorphism of this space that maps each orbit of one onto an orbit of the other and preserves the time orientation. The topological entropy (and also the measure-theoretic entropy) of a flow are defined using its time one map, see [3] [7] and [21]. Then, due to possible reparametrizations of orbits, these entropies may assume different values in a class of topological equivalence. How- ever, if a flow has no fixed points and its topological entropy is 0 or ∞ then it is 2000 Mathematics Subject Classification. MSC: 37C15, 34C28, 37A10. Key words and phrases. Equivalent flows, topological entropy, measure-theoretic entropy, reparametrizations. *Partially supported by FAPESP (# 2008/01877-5) and NSFC (# 10671006) and NBRPC(973 Program)(# 2006CB805900). **Partially supported by CNPq (# 304517/2005-4). This work is part of the “Projeto Tem´ atico - Fapesp # 2006/03829-2”. 121