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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)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,. Moreover, J contains an uncountable subset S devoid of asymptotically periodic points, such that

0=lim inf|F n (q)-F n (r)|<lim sup|F n (q)-F n (r)|

for all qr 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

92D25Population dynamics (general)
39A10Additive difference equations
54H20Topological dynamics
37N25Dynamical systems in biology
37C25Fixed points, periodic points, fixed-point index theory