  This paper is devoted to an in-depth study of the limiting measure of
Lee--Yang zeroes for the Ising Model on the Cayley Tree. We build on previous
works of M\"uller-Hartmann-Zittartz (1974 and 1977), Barata--Marchetti (1997),
and Barata--Goldbaum (2001), to determine the support of the limiting measure,
prove that the limiting measure is not absolutely continuous with respect to
Lebesgue measure, and determine the pointwise dimension of the measure at
Lebesgue a.e. point on the unit circle and every temperature. The latter is
related to the critical exponents for the phase transitions in the model as one
crosses the unit circle at Lebesgue a.e. point, providing a global version of
the "phase transition of continuous order" discovered by
M\"uller-Hartmann-Zittartz. The key techniques are from dynamical systems
because there is an explicit formula for the Lee-Yang zeros of the finite
Cayley Tree of level $n$ in terms of the $n$-th iterate of an expanding
Blaschke Product. A subtlety arises because the conjugacies between Blaschke
Products at different parameter values are not absolutely continuous.
