Michael Benedicks: Almost sure continuity along curves traversing the Mandelbrot set

Abstract. We study continuity properties of dynamical quantities while crossing the Mandelbrot set through typical smooth curves. In particular, we prove that for almost every parameter \(c_0\) in the boundary of the Mandelbrot set \(M\) with respect of the harmonic measure and every smooth curve   \(\gamma:[-1,1]\mapsto {\mathbb C}\)  with the property that \(c_0=\gamma(0)\)  there exists a set \({\mathcal A_\gamma}\)  having \(0\) as a Lebesgue density point and such that that \(\lim_{x\to 0}\mathop{\rm HDim}(J_{\gamma(x)}) =\mathop{\rm HDim}(J_{c_0})\) for the Julia sets \(J_c\).