Abstract
For pt.II see ibid. vol.17, p.435 (1984). In the first two papers of this series the authors considered self-similar fractal lattices with a finite order of ramification R. In the present paper they study physical models defined on a family of fractals with R= infinity . In order to characterise the geometry of these systems, they need the connectivity Q and the lacunarity L, in addition to the fractal dimensionality D. It is found that discrete-symmetry spin models on these lattices undergo a phase transition at Tc>0. An approximate renormalisation group scheme is constructed and used to find the dependence of Tc and the critical exponents on the geometrical factors. They also solve the problem of resistor networks on these fractals, and discuss its consequences concerning spin models with continuous symmetry.