Based on joint works with Guus Regts, and with Ferenc Bencs, Pjotr Buys and Lorenzo Guerini.

We consider the partition function of the Ising Model for graphs of bounded degree. The set of parameters for which this partition function can be zero is of importance both from a physics and from a computational complexity perspective. Our motivation here is the latter, focused on the existence of polynomial time algorithms for approximating the Ising partition function. Through the works of Barvinok and Patel&Regts it is known that such algorithms exist on the maximal open set (containing the origin) that contains no zero parameters.

The link with dynamical systems arises when considering graphs that are recursively defined. The simplest example is the set of perfect d-ary trees. For these trees the zero-parameters can be studied through iteration of a one-parameter family of rational functions of degree d. More generally one can consider the class of spherically regular trees, which induce a semi-group action, generated by the same rational functions of degree 1 through d.

The semi-group contains of course the iterates of the single rational function, corresponding to the inclusion of the set of perfect d-ary trees in the set of spherically regular trees. One would therefore expect that the set of zero-parameters of the larger class is much larger than that of the smaller class. It turns out that this is not the case for the ferromagnetic Ising Model: the closures of the two sets of zero-parameters are equal. However, equality fails for the anti-ferromagnetic Ising Model: in the relevant region the smaller set of zero-parameters is a Cantor set, while the larger contains a continuum. An important step in proving the existence of this continuum consists of proving that the sequences in the semi-group act uniformly hyperbolically on their respective Julia sets.