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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 $$F^k(p) = p$$. If $$p$$ has period 3 then the relation $$F^3(q)\leq q < F(q) < F^2(q)$$ (or its reverse) is satisfied for $$q$$ one of the points $$p$$, $$F(p)$$, or $$F^2(p)$$. 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 $$K=1,2,3,\dots$$. Moreover, $$J$$ contains an uncountable subset $$S$$ devoid of asymptotically periodic points, such that $0=\liminf|F^n(q)-F^n(r)| < \limsup|F^n(q)-F^n(r)|$ for all $$q\neq r$$ in $$S$$. (a point is asymptotically periodic if $$\lim|F^n(p) - F^n(q)| = 0$$ 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 $$x_{n+1} = F(x_n) = rx_n(1-x_n/K)$$ with a model of which, by contrast, $$|dF(x)/dx|>1$$ wherever the derivative exists. For such a system no periodic point is stable, in the sense that $$|F^k(q)-p| < |q-p|$$ 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 (MSC2010) 37N25 Dynamical systems in biology 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
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