Recurrence, symbolic dynamics, and wild attractors for unimodal maps

Lori Alvin; Jernej Činč

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arXiv · 2606.02308

Recurrence, symbolic dynamics, and wild attractors for unimodal maps

Lori Alvin·Jernej Činč

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Abstract

In this paper, we construct families of unimodal maps with wild Cantor attractors and study, via symbolic dynamics, the interplay between recurrence properties of the critical orbit. Using kneading and co-kneading techniques, we provide a symbolic characterization of persistent recurrence in terms of cutting and co-cutting times, and relate it to other recurrence conditions, including the Collet--Eckmann condition. As an application, we obtain uncountable families of non-renormalizable unimodal maps of sufficiently high critical order admitting wild attractors. We also analyze the structure of postcritical dynamics in recurrent regimes, constructing examples with embedded odometers and with minimal homeomorphic dynamics on the critical $\omega$-limit set, and showing that a minimal homeomorphism together with regular recurrence does not imply conjugacy to an adding machine. Finally, we prove that longbranched combinatorics is incompatible with persistent recurrence, thereby clarifying the boundary between different recurrence regimes.

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Lori Alvin, Jernej Činč. 2026-06-01. Recurrence, symbolic dynamics, and wild attractors for unimodal maps. https://arxiv.org/abs/2606.02308

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