Products of random matrices and derivatives on p.c.f. fractals

We define and study intrinsic first order derivatives on post critically finite fractals and prove differentiability almost everywhere with respect to self-similar measures for certain classes of fractals and functions. We apply our results to extend the geography is destiny principle to these cases, and also obtain results on the pointwise behavior of local eccentricities on the Sierpiński gasket, previously studied by Öberg, Strichartz and Yingst, and the authors. We also establish the relation of the derivatives to the tangents and gradients previously studied by Strichartz and the authors. Our main tool is the Furstenberg-Kesten theory of products of random matrices.