Positive Lyapunov exponent for generic one-parameter families of unimodal maps
The aim of this paper is to generalize results of \textit{M. Benedicks} and \textit{L. Carleson} [Ann. Math. (2) 133, No. 1, 73--169 (1991; Zbl 0724.58042)] concerning the dynamics of quadratic maps \(q_ a = 1 - ax^ 2\), and of the Hénon map. The conditions and results presented are highly technical, so we limit this review to a statement of a special case of the main theorem. Theorem. Let \(a^*\) be a parameter value such that: (i) \(f_{a^*}\) is a \({\mathcal C}^ 3\)-unimodal map, with negative Schwarzian derivative and \(f^{\prime \prime}_{a^*} (0) \neq 0\), (ii) there exist a constant \(N \geq 1\) such that \(c_ N(a^*)\) is a non-stable periodic point \(x^*\) of period \(p\), (iii) if \([a \to \chi(a)]\) denotes a local smooth continuation of \(x^*\), then \[ {d\over da} (\chi - c_ N) (a^*) \neq 0. \] Then \(a^*\) is a Lebesgue density point of a set of parameters \({\mathcal A}_{K,\lambda}\) of positive Lebesgue measure for some constants \(K\), \(\lambda\) such that for all \(a \in {\mathcal A}_{K,\lambda}\): (a) for all \(n \geq 0\), \(| Df^ n_ a (c_ 1(a))| \geq K \exp (n\lambda)\), (b) \(f_ a\) admits an invariant probability measure absolutely continuous with respect to Lebesgue measure.
scientific article; zbMATH DE number 718518
scientific article