an:03547131 Zbl 0351.92021 Li, Tien-Yien; Yorke, James A. Period three implies chaos EN Am. Math. Mon. 82, 985-992 (1975). 00140141 1975
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92D25 39A10 54H20 37N25 37C25 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. G.K. Francis