Multi-separation, centrifugality and centripetality imply chaos

Abstract:

Let $I$ be an interval. $I$ need not be compact or bounded. Let $f:I\rightarrow I$ be a continuous map, and $(x_0, x_1, \cdots , x_n)$ be a trajectory of $f$ with $x_n\leq x_0<x_1$ or $x_1<x_0\leq x_n$. Then there is a point $v\in I$ such that $\min \{x_0, \cdots , x_n\}<v=f(v)<\max \{x_0, \cdots , x_n\}$. A point $y\in I$ 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 $\{x_0, \cdots , x_{n-1}\}, k\geq 1$, then $f$ has periodic points of some odd ($\not = 1$) period $p\leq (n-2)/k+2$. In addition, we also prove that if $(x_0, x_1, \cdots , x_{n-1}$) is multi-separated by Fix($f$), or there exists a centrifugal point of $f$ in $\{x_0, \cdots , x_{n-1}\}$, then $f$ is turbulent and hence has periodic points of all periods.