Abstract

$\tau$ is a continuous map on a metric compact space $X$. For a continuous function $\phi:X\to\mathbb R$ we considera 1-dimensional map $T$ (possibly multi-valued) which sends a local $\phi$-maximum on $\tau$ 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 $\tau$ is in some sense chaotic, i.e., it has a turbulent trajectory and thus a continuous invariant measure.