Topological entropies of equivalent smooth flows

TL;DR

This paper constructs two equivalent smooth flows with a singularity that have different topological entropies, showing that zero entropy is not an invariant for differentiable flows.

Contribution

It provides a counterexample of equivalent smooth flows with different entropies, answering Ohno's question negatively in the class of $C^ abla$ flows.

Findings

Constructed two equivalent $C^ abla$ smooth flows with different entropies

Showed zero topological entropy is not invariant under flow equivalence in smooth category

Extended understanding of entropy invariance for smooth dynamical systems

Abstract

Two flows defined 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, finite 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 fixed points there exists a counterexample, constructed by Ohno, showing that neither 0 nor $\infty$ 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 $\infty$ entropy. These facts led Ohno in 1980 to ask the following: "Is 0 topological entropy an invariant for equivalent differentiable flows?" In this paper, 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 in the class $C^\infty$.