Hölder exponent of planar Julia sets associated with polynomials

If the Green function gE of a compact set E ⊂ C is Holder continuous, then the Holder exponent of the set E is the supremum over all such α, that |gE(z)− gE(w)| ≤M |z −w|α, z, w ∈ C. We give a lower bound for the Holder exponent of the Julia sets of polynomials. In particular we show that there are totally disconnected planar sets with the Holder exponent greater than 3/5.