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Period 3 and chaos for unimodal maps

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  • In this paper we study unimodal maps on the closed unit interval, which have a stable period 3 orbit and an unstable period 3 orbit, and give conditions under which all points in the open unit interval are either asymptotic to the stable period 3 orbit or land after a finite time on an invariant Cantor set $\Lambda$ on which the dynamics is conjugate to a subshift of finite type and is, in fact, chaotic. For the particular value of $\mu=3.839$, Devaney [3], following ideas of Smale and Williams, shows that the logistic map $f(x)=\mu x(1-x)$ has this property. In this case the stable and unstable period 3 orbits appear when $\mu=\mu_0=1+\sqrt{8}$. We use our theorem to show that the property holds for all values of $\mu>\mu_0$ for which the stable period 3 orbit persists.
    Mathematics Subject Classification: Primary: 37E05, 37D05; Secondary: 37E15.

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