Regular or stochastic dynamics in real analytic families of unimodal maps
Arguably, the most remarkable achievement in interval dynamics during the last quarter of 20th century has been the unravelling of the ``a.e. point-a.e. parameter'' dynamical structure of the quadratic family \(q_\tau(x)=\tau-1-\tau x^2\), \(\tau\in [1/2,2]\). This fruitful trend of work starts with \textit{J. Guckenheimer} [Commun. Math. Phys. 70, 133--160 (1979; Zbl 0429.58012)] and \textit{M. V. Jakobson} [Commun. Math. Phys. 81, 39--88 (1981; Zbl 0497.58017)] and culminates with \textit{M. Lyubich} [Acta Math. 178, 185--297 (1997; Zbl 0908.58053)], \textit{J. Graczyk} and \textit{G. Swiątek} [Ann. Math. (2) 146, 1--52 (1997; Zbl 0936.37015)] and \textit{M. Lyubich} [Ann. Math. (2) 156, 1--78 (2002; Zbl 1160.37356)], and has succeeded in proving that the set of parameters \(\tau\) such that \(q_\tau\) is regular (this means that all periodic orbits of the map are hyperbolic and there is a periodic orbit attracting Lebesgue almost all \(x\in [-1,1]\)) is open and dense in \([1/2,2]\), and the set of parameters \(\tau\) such that \(q_\tau\) is stochastic (which means that the map admits an invariant measure absolutely continuous with respect to the Lebesgue measure) has positive Lebesgue measure and, moreover, full measure in the complementary of regular parameters. Hence, in particular, almost every quadratic map is either regular or stochastic. In the paper under review, the latter result is extended to many analytic families of analytic unimodal maps (including all nontrivial analytic families of unimodal maps with negative Schwarzian derivative). To this aim the space of analytic maps is foliated by codimension-one analytic submanifolds (so-called ``hybrid classes''), each containing one quadratic map \(q_\tau\), which enables the authors to export the regular-stochastic dichotomy of the quadratic family to other analytic families. In order to state precisely this result, we need a number of definitions. Let \(\widetilde{\mathcal{U}}^3\) denote the family of \(C^3\)-unimodal maps \(f:[-1,1]\rightarrow [-1,1]\) satisfying \(f(-1)=f(1)=-1\), \(f'(-1)\geq 1\) and \(f''(c)<0\), with \(c\) being the turning point of \(f\). A map \(f\in\widetilde{\mathcal{U}}^3\) is said to be quasi-quadratic if any nearby map \(g\in \widetilde{\mathcal{U}}^3\) is topologically conjugate to some map of the quadratic family. It must be emphasized that the set \(\widetilde{\mathcal{U}}\) of quasi-quadratic maps includes all maps from \(\widetilde{\mathcal{U}}^3\) having negative Schwarzian derivative. If \(a>0\) and \(\Omega_a=\{z\in \mathbb{C}: \text{dist}(z,I)<a\}\) (\(I=\{(x,0):x\in [-1,1]\}\)) then \(\widetilde{\mathcal{U}}_a\) denotes the family of quasi-quadratic real analytic maps which can be analytically and continuously extended as complex maps to, respectively, \(\Omega_a\) and \(\text{Cl}\,\Omega_a\). We say that two quasi-quadratic maps belong to the same hybrid class if they have the same kneading invariant and the multipliers of corresponding nonrepelling periodic points are the same for both maps (if \(p\) is a periodic point of \(f\) of period \(r\) then its multiplier is \((f^r)'(p)\)). We say that a family of quasi-quadratic maps is nontrivial if it is not contained in a single hybrid class. Theorem: Let \(\{f_\lambda\}_{\lambda\in \Lambda} \subset \widetilde{\mathcal{U}}_a\) be a nontrivial one-parameter real analytic family of quasiquadratic maps. Then for Lebesgue almost all parameters \(\lambda\), the map \(f_\lambda\) is either regular or stochastic. This impressive paper has been further generalized by \textit{A. Avila} and \textit{C. G. Moreira} [Phase-parameter relation and sharp statistical properties for general families of unimodal maps (preprint)].
scientific article; zbMATH DE number 2078254
scientific article