Abstract:
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.
