Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials

For an infinitely renormalizable quadratic map f(c) : z -> z(2) + c with the sequence of renormalization periods {k (m) } and rotation numbers {t (m) = p (m) /q (m) }, we prove that if lim sup k(m)(-1) log vertical bar p(m)vertical bar > 0, then the Mandelbrot set is locally connected at c. We prove also that if lim sup vertical bar t(m+1)vertical bar(1/qm) < 1 and q (m) -> a, then the Julia set of f (c) is not locally connected and the Mandelbrot set is locally connected at c provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed by J. Milnor.