Summability implies Collet-Eckmann almost surely
The main result of this paper is a generalization of the famous theorem of \textit{M. Jakobson} [Commun. Math. Phys. 81, 39--88 (1981; Zbl 0497.58017)]. For a family of interval maps \(\{f_t\}\) such that \(f_0\) satisfies a summability condition, the authors prove that \(t=0\) is a Lebesgue density point of parameters corresponding to maps satisfying the Collet-Eckmann condition and the strong polynomial recurrence condition.NEWLINENEWLINEThis is done using adapted results of \textit{W. Shen} [Proc. Lond. Math. Soc. (3) 107, No. 5, 1091--1134 (2013; Zbl 1290.37022)] and \textit{M. Tsujii} [Invent. Math. 111, No. 1, 113--137 (1993; Zbl 0787.58029)]. Through a theorem of \textit{G. Levin} [``Perturbations of weakly expanding critical orbits'', in: Frontiers in complex dynamics: in celebration of John Milnor's 80th birthday. Princeton: Princeton University Press. 163--196 (2014)], the main result is extended to higher dimensions.
scientific article; zbMATH DE number 6359409
scientific article