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 Müller-Hartmann-Zittartz (1974 and, 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 Müller-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.

Associated to any finite simple graph Γ is the chromatic polynomial PΓ(q) whose complex zeros are called the chromatic zeros of Γ. A hierarchical lattice is a sequence of finite simple graphs {Γn} ∞ n=0 built recursively using a substitution rule expressed in terms of a generating graph. For each n, let µn denote the probability measure that assigns a Dirac measure to each chromatic zero of Γn. Under a mild hypothesis on the generating graph, we prove that the sequence µn converges to some measure µ as n tends to infinity. We call µ the limiting measure of chromatic zeros associated to {Γn} ∞ n=0 . In the case of the Diamond Hierarchical Lattice we prove that the support of µ has Hausdorff dimension two.The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications.Date: March 23, 2021.1 For example, a search on Mathscinet yields 333 papers having the words "chromatic polynomial" in the title.

Associated to any finite simple graph Γ is the chromatic polynomial PΓ(q) whose complex zeros are called the chromatic zeros of Γ. A hierarchical lattice is a sequence of finite simple graphs {Γn} ∞ n=0 built recursively using a substitution rule expressed in terms of a generating graph. For each n, let µn denote the probability measure that assigns a Dirac measure to each chromatic zero of Γn. Under a mild hypothesis on the generating graph, we prove that the sequence µn converges to some measure µ as n tends to infinity. We call µ the limiting measure of chromatic zeros associated to {Γn} ∞ n=0 . In the case of the Diamond Hierarchical Lattice we prove that the support of µ has Hausdorff dimension two.The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications.

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