We consider a family of discrete Jacobi operators on the one-dimensional integer lattice
with Laplacian and potential terms modulated by a primitive invertible two-letter
substitution. We investigate the spectrum and the spectral type, the fractal structure and
fractal dimensions of the spectrum, exact dimensionality of the integrated density of
states, and the gap structure. We present a review of previous results, some applications,
and open problems. Our investigation is based largely on the dynamics of trace maps. This
work is an extension of similar results on Schrödinger operators, although some of the
results that we obtain differ qualitatively and quantitatively from those for the
Schrödinger operators. The nontrivialities of this extension lie in the dynamics of the
associated trace map as one attempts to extend the trace map formalism from the
Schrödinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are,
in a sense, a test item, as many other models can be attacked via the same techniques, and
we present an extensive discussion on this.