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arXiv · 2608.10655

Absolutely continuous invariant measures and the Collet-Eckmann condition for long-branched Sturmian unimodal maps

Jorge Olivares-Vinales
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Abstract

We study long-branched unimodal maps for which the sequence of values of the kneading map form a Sturmian word. We identify these kneading classes with those arising from the stunted Lorenz construction of Anu\v{s}i\'c, Bruin, and \v{C}in\v{c}, and their quadratic representatives with the maps constructed by Bl\'e from rigid rotations. We compute the cutting and co-cutting times explicitly and establish criteria for the existence and nonexistence of absolutely continuous invariant probability measures. We also show that, for Lebesgue-almost every rotation angle, no finite-order $S$-unimodal realization satisfies the Collet-Eckmann condition.

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Jorge Olivares-Vinales. 2026-08-11. Absolutely continuous invariant measures and the Collet-Eckmann condition for long-branched Sturmian unimodal maps. https://arxiv.org/abs/2608.10655

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