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Intermittency in families of unimodal maps. (English) Zbl 1032.37020

Consider a one parameter family \(\{f_{\gamma}\}\) of unimodal maps with negative Schwarzian derivative. Suppose that for \(\gamma =0\) we have a boundary crisis bifurcation or saddle node bifurcation. This means, roughly spoken, that in a neighbourhood of \(0\) in the parameter space, \(f_{\gamma}\) has a periodic interval, respectively a periodic orbit, for \(\gamma <0\), which disappears for \(\gamma >0\). Very roughly spoken, there is a set \(N\), which is an attractor for \(\gamma\leq 0\), and this attractor disappears for \(\gamma >0\). For \(x\) and \(\gamma\) define \(\Phi (x,\gamma)=\lim_{n\to\infty}\frac{1}{n} \sum_{j=0}^{n-1}1_{N}(f_{\gamma}^{j}(x))\) (this means the frequency with which the orbit of \(x\) visits \(N\)). It is proved that for every rational number \(q\in (0,1)\) and every \(\varepsilon >0\) there exists \(0<\gamma <\varepsilon\) such that \(\Phi (x,\gamma)=q\) for almost all \(x\). On the other hand, there exists a set \(\Gamma\) of parameter values having positive Lebesgue measure and positive density at \(0\) such that for \(\gamma\in\Gamma\) the value \(\Phi (x,\gamma)\) is constant almost everywhere. Denoting this constant by \(\Phi (\gamma)\) one obtains that \(\Phi (\gamma)\) is continuous at \(\gamma=0\) (if considered only on \(\Gamma\)).
Although sometimes it is hard to read because of the (necessary) technical details, this well written papers deserves to be read.
Reviewer: Peter Raith (Wien)

MSC:

37E05 Dynamical systems involving maps of the interval
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37G99 Local and nonlocal bifurcation theory for dynamical systems
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