2022/11/14 by Bogdanov, Konstantin, Bufetov, Alexander
#37E05 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2211.07252
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 θ∈(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 θ are bounded (Theorem 1.2) or, in the case of quadratic polynomials, have sufficiently slow growth (Theorem 1.3).