Multi–separation, Centrifugality and Centripetality Imply Chaos

Let I be an interval. I need not be compact or bounded. Let f : I → I be a continuous map, and (x0, x1, · · · , xn) be a trajectory of f with xn ≤ x0 < x1 or x1 < x0 ≤ xn. Then there is a point v ∈ I such that min{x0, · · · , xn} < v = f(v) < max{x0, · · · , xn}. A point y ∈ I is called a centripetal point of f relative to v if y < f(y) < v or v < f(y) < y, and y is centrifugal if f(y) < y < v or v < y < f(y). In this paper we prove that if there exist k centripetal points of f in {x0, · · · , xn−1}, k ≥ 1, then f has periodic points of some odd (6= 1) period p ≤ (n−2)/k+2. In addition, we also prove that if (x0, x1, · · · , xn−1) is multi-separated by Fix(f), or there exists a centrifugal point of f in {x0, · · · , xn−1}, then f is turbulent and hence has periodic points of all periods.