Hölder exponents of the Green functions of planar polynomial Julia sets

Abstract

If the Green function gE of a compact set \({E \subset \mathbb{C}}\) is Holder continuous, then the Holder exponent of the set E is the supremum over all such α that $$|g_E(z)-g_E(w)|\leq M|z-w|^\alpha,\, z, w \in \mathbb{C}.$$ We give a lower bound for the Holder exponent of the Julia sets of polynomials. In particular, we show that there exist totally disconnected planar sets with the Holder exponent greater than 1/2 as well as fat continua with the boundary nowhere smooth and with the Holder exponent as close to 1 as we wish.