Colloquium

Abstract: We construct two equivalent smooth flows, one of which has positive topological entropy and the other has zero topological entropy. Two flows defined on a smooth manifold are equivalent if there exists a homeomorphism of the manifold that sends each orbit of one flow onto an orbit of the other flow while preserving the time orientation. The topological entropy of a flow is defined as the entropy of its time-1 map. While topological entropy is an invariant for equivalent homeomorphisms, finite non-zero topological entropy for a flow cannot be an invariant because its value is affected by time reparameterization. However, 0 and \(\infty\) topological entropy are invariants for equivalent flows without fixed points.

In equivalent flows with fixed points there exists a counterexample, constructed by Ohno, showing that neither 0 nor infinite topological entropy is preserved by equivalence. The two flows constructed by Ohno are suspensions of a transitive subshift and thus are not differentiable. Note that a differentiable flow on a compact manifold cannot have infinite entropy. These facts led Ohno in 1980 to ask the following: Is 0 topological entropy an invariant for equivalent differentiable flows?

In this talk, we construct two equivalent \(C^\infty\) smooth flows with a singularity, one of which has positive topological entropy while the other has zero topological entropy. This gives a negative answer to Ohno's question.

The key idea used by Ohno and in this work is that the time reparameterization between an orbit in one flow and its image orbit in an equivalent flow can grow super-exponentially near a fixed point. In the present case we construct the two flows as suspensions of an example by Herman of a diffeomorphism that is minimal (every forward orbit is dense) and has positive topological entropy.

This work is joint with Sun Wenxiang and Zhou Yunhua.