DYNAMICAL-SYSTEMS FOR QUASI-PERIODIC CHAINS AND NEW SELF-SIMILAR POLYNOMIALS

Dynamical systems in SL(2, R) or SL(2, C) naturally appear in the tran sfer matrix method for quasiperiodic chains characterized by arbitrary irrational numbers. We show new sub-dynamical systems and invariants that are related to full diagonal and off-diagonal components of the t ransfer matrices; they are analogous to formulae of Chebyshev polynomi als of the first and second kinds. Applying them to an electronic prob lem on the Fibonacci chain, we obtain sets of self-similar polynomials , quasiperiodic extension of the Chebyshev polynomials of the first an d second kinds with self-similar properties. Two scaling factors of th e self-similarities coincide with ones obtained by the perturbative de cimation renormalization group method.