We have studied quadric type energy (frequency) recursion relations resulting from fractal lattices. We found that the allowed energy intervals in successive levels form two similar two-scale Cantor sets. If WE imagine the iterating procedure as a dynamical process, the iterating results in different levels gen-erate a"time"sequence, and we can introduce an equal probability measure Pu on a Cantor set and con-struct，following Helsey et a1. [Phys. Rev. A 33, 1141 (1986)], a partition function F (q,T) =∑ipq/17; finally, we obtain Dq-q and f (a)-a curves. A number of exactly soluable examples are investigated.