Chaos for successive maxima map implies chaos for the original map

τ is a continuous map on a metric compact space X . For a continuous function ϕ : X → R , we consider a one-dimensional map T (possibly multi-valued) which sends a local ϕ -maximum on τ trajectory to the next one: consecutive maxima map. The idea originated with famous Lorenz’s paper on strange attractor. We prove that if T has a horseshoe disjoint from fixed points, then τ is in some sense chaotic, i.e., it has a turbulent trajectory and thus a continuous invariant measure.