## Institute for Mathematical Sciences

## Preprint ims90-12

** P. M. Bleher & M. Lyubich**
* The Julia Sets and Complex Singularities in Hierarchical Ising Models*

Abstract: We study the analytical continuation in the complex plane of free energy of the Ising model on diamond-like hierarchical
lattices. It is known that the singularities of free energy of
this model lie on the Julia set of some rational endomorphism
$f$ related to the action of the Migdal-Kadanoff renorm-group.
We study the asymptotics of free energy when temperature goes
along hyperbolic geodesics to the boundary of an attractive
basin of $f$. We prove that for almost all (with respect to
the harmonic measure) geodesics the complex critical exponent
is common, and compute it.

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