The author considers more or less the class of p.c.f. self-similar fractals. A potential theory on such sets requires for example a “Dirichlet integral” or, more precisely, its abstract relative, a Dirichlet form. The existence and uniqueness of such forms can be traced back to a finite dimensional nonlinear fixed point problem for the so called renormalization map. The idea is that the scaling property of the Dirichlet form should be adapted to the self-similarity of the underlying fractal. The main results are:
Corollary 5.7: Every strongly symmetric fractal has at most one Dirichlet form.
Theorem 4.22: Whenever there exists a Dirichlet form then it can be approximated by iterating the renormalization map.
The major technical ingredients are, firstly, Perron-Frobenius theory for inhomogeneous products of nonnegative matrices to obtain weak ergodicity and, secondly, Hilbert’s projective metric on cones, that is, nonlinear “Perron-Frobenius theory”, to turn weak ergodicity into strong ergodicity.