We prove that for every polynomial of one complex variable of degree at least
2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdor
dimension of this Julia set is larger than 1. Till now this was known only in the
connected Julia set case.
We give also an (easy) example of a polynomial with non-connected Julia set and
all non one-point components being analytic arcs, thus contradicting Ch. Bishop's
conjecture that such components must have Hausdor dimension larger than 1.
This is a joint work with Feliks Przytycki.