Monday 09 Nov 2015: Almost sure continuity along curves traversing the Mandelbrot set

Michael Benedicks - KTH Stockholm

*** Harrison 209 *** 14:30-15:30

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 $\lim_{x\to 0} HDim(J_{\gamma(x)} =HDim(J_{c_0})$ for the Julia sets $J_c$. (joint work with Jacek Graczyk)