We explore the dynamics of the two by two projective general linear group with coefficients in a field of formal series over a finite field, focusing on the right translation of diagonal elements and unipotent elements on the quotient by the modular group whose coefficients are polynomials over the same finite field. The correspondence between the dynamics on the space of cosets and the set of ordered triple points will be given. We also discuss some applications to number theory as well as the effective rigidity of the Haar measure.