  Trace maps of two-letter substitution rules are investigated with special
emphasis on the underlying algebraic structure and on the existence of
invariants. We illustrate the results with the generalized
  Fibonacci chains and show that the well-known Fricke character
  I(x,y,z) = x^2 + y^2 + z^2 - 2 x y z - 1 is not the only type of invariant
that can occur. We discuss several physical applications to electronic spectra
including the gap-labeling theorem, to kicked two-level systems, and to the
classical 1D Ising model with non-commuting transfer matrices.
