Since the discovery of quasicrystals by Schechtman et. al.
in 1984, quasi-periodic models in mathematical physics have formed an
active area of research. In particular, effects of quasi-periodicity
were investigated in a widely studied model of magnetism: the Ising
model (quantum and classical). Numerical and some analytic results
began to appear in the late '80s; however, most interesting
(numerical) results hitherto remained rigorously unconfirmed.Most of
the previous results relied on a connection with hyperbolic dynamical
systems.It is our aim to rigorously confirm previous numerical
observations, as well as to prove new results, by exploiting further
the aforementioned connection. In particular, we'll prove
multi-fractal structure of the energy spectrum of one-dimensional
quantum quasi-periodic Ising models. We'll also discuss its fractal
dimensions and measure.