Absolute continuity of non-homogeneous Gibbs measures of the Ising model on the Cayley tree

In this paper, for the Ising model on the Cayley tree of order k ⩾ 2 , a sequence { h n } of boundary conditions is constructed depending on an initial value h which defines a Gibbs measure µ _h . By investigating the dynamical behaviour of the renormalisation group map associated with the model, we prove that each measure µ _h is equivalent to the disordered phase μ ∗ . This result shines a new light to the question closely related to the classical result by Kakutani which asserts that any two locally-equivalent probability product measures are either equivalent or mutually-singular.