Theory of quasiperiodic lattices. I. Scaling transformation for a quasiperiodic lattice

A theory of quasiperiodic lattices that consist of only two types of atoms is introduced. If a ratio between the numbers of the two types of atoms in a chain is irrational, the lattice is quasiperiodic. First, we present the scaling transformation for the construction of any quasiperiodic lattice in terms of a continued-fraction expansion of the ratio. Second, we show that all the scaling transformations preserve an invariant surface, which was first discovered by Kohmoto, Kadanoff, and Tang.