Topological entropies of equivalent smooth flows

Abstract

Abstract. We construct two equivalent smooth flows, one of which has positive topological entropy and the other has zero topological entropy. This provides a negative answer to a problem posed by Ohno.

A countable partition for singular flows, and its application on the entropy theory

“…It is a well accepted fact that for flows with singularities, the topological entropy, as well as metric entropies, can behave in a rather bizarre way. For example, in [23] the authors constructed C ∞ equivalent flows, such that one has zero entropy while the other has positive entropy. Even with those cross sections in [8] and [18], the unbounded return time, which results in the unbounded derivative for the return map, has been proven to be the main obstruction for the ergodic theory of singular flows.…”

Distributional chaos for flows

“…P r o o f. We will construct two equivalent flows ψ and ϕ satisfying the theorem. The essential construction is included in [6] and we mainly prove that this is suitable for our goal.…”

“…By [6], there are two (smooth) flows ψ and ϕ on a compact space M (in fact, M is a smooth manifold) such that ψ is a time-changed flow of ϕ (and hence ψ is equivalent to ϕ) and both flows have only the same singular point p. Furthermore, h(ψ) > 0 and the Dirac measure µ p is the only invariant measure of ϕ. Then, by Corollary 2.1, ψ is DC2 since h(ψ) > 0.…”

Weak Horseshoe with bounded-gap-hitting times

“…In [10], Ohno constructed a counterexample of equivalent flows with fixed points to indicate that neither zero nor infinity topological entropy is preserved by equivalence. Furthermore, in [12], the authors proved that zero topological entropy is not an invariant of equivalent differentiable flow. However, we do not know whether weak horseshoe with boundedgap-hitting times is an invariant of equivalent flows.…”