  In the present paper, by conducting research on the dynamics of the $p$-adic
generalized Ising mapping corresponding to renormalization group associated
with the $p$-adic Ising-Vannemenus model on a Cayley tree, we have determined
the existence of the fixed points of a given function. Simultaneously, the
attractors of the dynamical system have been found. We have come to a
conclusion that the considered mapping is topologically conjugate to the
symbolic shift which implies its chaoticity and as an application, we have
established the existence of periodic $p$-adic Gibbs measures for the $p$-adic
Ising-Vannemenus model.

Dynamical Systems