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arXiv · 2211.07252

Irrational rotation dynamics for unimodal maps

Konstantin Bogdanov·Alexander Bufetov
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Abstract

The first result of the paper (Theorem 1.1) is an explicit construction of unimodal maps that are semiconjugate, on the post-critical set, to the circle rotation by an arbitrary irrational angle $\theta\in(3/5,2/3)$. Our construction is a generalization of the construction by Milnor and Lyubich [LM] of the Fibonacci unimodal maps semi-conjugate to the circle rotation by the golden ratio. Generalizing a theorem by Milnor and Lyubich for the Fibonacci map, we prove that the Hausdorff dimension of the post-critical set of our unimodal maps is $0$, provided the denominators of the continued fraction of $\theta$ are bounded (Theorem 1.2) or, in the case of quadratic polynomials, have sufficiently slow growth (Theorem 1.3).

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Konstantin Bogdanov, Alexander Bufetov. 2022-11-14. Irrational rotation dynamics for unimodal maps. https://arxiv.org/abs/2211.07252

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