An entropic tour of the Mandelbrot set

A fundamental theme in holomorphic dynamics is that the local geometry of parameter space (e.g. the Mandelbrot set M) near a parameter reflects the geometry of the Julia set, hence ultimately the dynamical properties, of the corresponding dynamical system. We shall discuss a new instance of this principle in terms of entropy. Indeed, recently W. Thurston defined the core entropy of the map f_c = z^2 + c as the entropy of the restriction of f_c to its Hubbard tree. The core entropy changes very interestingly as the parameter c changes, and we shall relate such variation to the geometry of M. Namely, we shall compare the Hausdorff dimension of certain sets of external rays landing on veins of M to the core entropy of quadratic polynomials f_c along the vein.