Irrational rotation dynamics for unimodal maps

TL;DR

This paper constructs specific unimodal maps that are semi-conjugate to irrational circle rotations, analyzes their post-critical sets, and establishes conditions under which these sets have Hausdorff dimension zero.

Contribution

It generalizes previous work by Milnor and Lyubich, providing explicit constructions for a broader class of irrational rotations and analyzing their fractal properties.

Findings

Constructed unimodal maps semi-conjugate to irrational rotations with angles in (3/5, 2/3)

Proved the Hausdorff dimension of the post-critical set is zero under certain conditions

Extended results to quadratic polynomials with slow-growing continued fraction denominators

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).