Hölder exponents of the Green functions of planar polynomial Julia sets (Q2014994)

The authors investigate compact planar sets \(E \subset \mathbb{C}\) with Hölder continuous Green functions \(g_E\), i.e., such that there exist positive constants \(M\) and \(\alpha\) with \(|g_E(z) - g_E(w)| \leq M|z - w|^{\alpha}\), for \(z, w \in \mathbb{C}\), and define the Hölder exponent of the set \(E\) to be \[ \lambda(E) = \text{sup} \big\{ \alpha: |g_E(z) - g_E(w)| \leq M|z - w|^{\alpha} \text{ holds with the exponent} \; \alpha \big\}. \] In the article the authors deal with Julia sets associated to a complex polynomial of degree \(d \geq 2\). They give a lower bound for the Hölder exponent of the Julia sets of polynomials. In particular, they show that there exist totally disconnected planar sets with Hölder exponent greater than 1/2 as well as fat continua with nowhere smooth boundary and with Hölder exponent arbitrary close to 1.