Topological chaos and statistical triviality

Abstract

While transferring one flow ϕ with fixed points homeomorphically to its equivalent flow ψ, we show that positive topological entropy degenerates to zero (such degeneracy happens for equivalent continuous flows [3] and equivalent differential flows [6]) if and only if all ergodic invariant measures with positive measure-theoretic entropy degenerate to fixed points. Whenever ϕ is assumed to be topological transitive, the measure degeneracy implies that the resulted equivalent flow ψ is topologically chaotic but statistically trivial, meaning that all ergodic invariant measures are supported on fixed points. Using different approaches in different areas people constructed examples of topological chaotic but statistical trivial systems, see [3] for C0 flows, see [6] for Cr, r≥1, flows, see [1][11] for C0 homeomorphisms, see time one map in [6] for Cr, r≥1 diffeomorphisms. We point out it is non-hyperbolic singularity causes the degeneracy while changing one flow equivalently to another.