Non-homogeneous Gibbs measures for the Ising model on the Cayley trees

In the present talk, we discuss the classical Ising model on the Cayley tree. For this model on the Cayley tree of order $k\geq 2$, a sequence $\{h_n\}$ of boundary conditions is constructed depending on an initial value $h$ which defines a Gibbs measure $\mu_h$. By investigating the dynamical behavior of the renormalization group map associated with the model, it will be discussed properties of each measure $\mu_h$. The obtained result is closely related to the classical result by Kakutani which asserts that any two locally-equivalent probability product measures are either equivalent or mutually-singular.