Dynamical systems for quasiperiodic chains and new self-similar polynomials

Toshi-kazu Suzuki and Tomonari Dotera

J. Phys. A: Math. Gen. 26 (1993), pp.6101-6113

Dynamical systems in SL(2, R) or SL(2, C) naturally appear in the transfer 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 transfer matrices; they are analogous to formulae of Chebyshev polynomials of the first and second kinds. Applying them to an electronic problem on the Fibonacci chain, we obtain sets of self-similar polynomials, quasiperiodic extension of the Chebyshev polynomials of the first and second kinds with self-similar properties. Two scaling factors of the self-similarities coincide with ones obtained by the perturbative decimation renormalization group method.