Renormalization-group techniques are applied to Ising-model spins placed on the sites of several self-similar fractal lattices. The resulting critical properties are shown to vary with the (noninteger) fractal dimensionality $D$, but also with several topological factors: ramification, connectivity, lacunarity, etc. For any $D>~1$, there exist systems with both $T_c=0\mathrm{}$, and $T_c>0\mathrm{}$; hence a lower critical dimensionality is not defined. The nonvanishing values of $T_c$ and the critical exponents depend on all these factors.

• Received 12 May 1980

DOI:https://doi.org/10.1103/PhysRevLett.45.855