## Institute for Mathematical Sciences

## Preprint ims10-04

** Pavel Bleher, Mikhail Lyubich, Roland Roeder**
*Lee-Yang zeros for DHL and 2D rational dynamics, I. Foliation of the physical cylinder*

Abstract: In a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising model always lie
on the unit circle. Distribution of the zeros is physically important
as it controls phase transitions in the model. We study this distribution
for the Migdal-Kadanoff Diamond Hierarchical Lattice (DHL). In this case,
it can be described in terms of the dynamics of an explicit rational
function R in two variables (the renormalization transformation). We
prove that R is partially hyperbolic on an invariant cylinder C. The
Lee-Yang zeros are organized in a transverse measure for the central-stable
foliation of R|C. Their distribution is absolutely continuous. Its density
is C^infty (and non-vanishing) below the critical temperature. Above the
critical temperature, it is C^infty on an open dense subset, but it vanishes
on the complementary Cantor set of positive measure. This seems to be the
first occasion of a complete rigorous description of the Lee-Yang
distributions beyond 1D models.

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