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Period three implies chaos. (English) Zbl 0351.92021

Let $F$ be a continuous function of an interval $J$ into itself. The period of a point in $J$ is the least integer $k>1$ for which ${F}^{k}\left(p\right)=p$. If $p$ has period 3 then the relation ${F}^{3}\left(q\right)\le q (or its reverse) is satisfied for $q$ one of the points $p$, $F\left(p\right)$, or ${F}^{2}\left(p\right)$. The title of the paper derives from the theorem that if some point $q$ in $J$ has this Sysiphusian feature, “two steps forward, one giant step back”, then $F$ has periodic points of every period $K=1,2,3,\cdots$. Moreover, $J$ contains an uncountable subset $S$ devoid of asymptotically periodic points, such that

$0=lim inf|{F}^{n}\left(q\right)-{F}^{n}\left(r\right)|

for all $q\ne r$ in $S$. (a point is asymptotically periodic if $lim|{F}^{n}\left(p\right)-{F}^{n}\left(q\right)|=0$ for some periodic point $p$.) The proof is eminently accessible to the nonspecialist and is therefore of interest to anyone modeling the evolution of a single population parameter by a first order difference equation. The authors compare the logistic ${x}_{n+1}=F\left({x}_{n}\right)=r{x}_{n}\left(1-{x}_{n}/K\right)$ with a model of which, by contrast, $|dF\left(x\right)/dx|>1$ wherever the derivative exists. For such a system no periodic point is stable, in the sense that $|{F}^{k}\left(q\right)-p|<|q-p|$ for all $q$ in a neigborhood of a periodic point $p$ of $k$. A brief survey of a theorem motivated by ergodic theory completes this fascinating paper.

Reviewer: G.K. Francis

MSC:
 92D25 Population dynamics (general) 39A10 Additive difference equations 54H20 Topological dynamics 37N25 Dynamical systems in biology 37C25 Fixed points, periodic points, fixed-point index theory