  In the present paper, we introduce a new kind of $p$-adic measures for
$q+1$-state Potts model, called {\it $p$-adic quasi Gibbs measure}. For such a
model, we derive a recursive relations with respect to boundary conditions.
Note that we consider two mode of interactions: ferromagnetic and
antiferromagnetic. In both cases, we investigate a phase transition phenomena
from the associated dynamical system point of view. Namely, using the derived
recursive relations we define one dimensional fractional $p$-adic dynamical
system. In ferromagnetic case, we establish that if $q$ is divisible by $p$,
then such a dynamical system has two repelling and one attractive fixed points.
We find basin of attraction of the fixed point. This allows us to describe all
solutions of the nonlinear recursive equations. Moreover, in that case there
exists the strong phase transition. If $q$ is not divisible by $p$, then the
fixed points are neutral, and this yields that the existence of the quasi phase
transition. In antiferromagnetic case, there are two attractive fixed points,
and we find basins of attraction of both fixed points, and describe solutions
of the nonlinear recursive equation. In this case, we prove the existence of a
quasi phase transition.
