One-dimensional generalized Fibonacci tilings

Abstract

Polynomial (nonsingular) dynamical trace maps of generalized Fibonacci tilings (A,B\ensuremath{\rightarrow}${\mathit{A}}^{\mathit{m}}$${\mathit{B}}^{\mathit{n}}$,A) are derived for arbitrary values of m and n. It is shown that these sequences can be grouped into two distinct classes. The sequences in class I correspond to n=1 and arbitrary m. They are shown to have volume-preserving and invertible trace maps with an invariant the same as that of the golden-mean sequence. The class-II sequences correspond to n>1 and arbitrary m and are shown to be associated with volume-nonpreserving and noninvertible trace maps with a common pseudoinvariant which is of the form of the invariant of class-I maps. Furthermore, it is shown for the class-II case that if n=m+1 the trace maps are two dimensional.