We study the
relationship between continued fraction expansions in *L*=F_{p}((*t*^{-1})),
the field of formal Laurent series in variable *t*^{-1}, and the
geodesic flow on the quotient of the projective linear group *G*=PGL_{2}(*L*)
by the integer lattice Γ=PGL_{2}(F_{p}[*t*]).
We review basic properties of continued fraction expansions and the continued
fraction map *T*(*x*)* =*{*x*^{-1}}. We discuss the
quotient of *G* by the maximal compact subgroup *K*=PGL_{2}(*O*),
where *O* is the closed unit interval in *L*. We visualize *G/K*
as the infinite *p*1 regular tree and use this visualization to study the
geodesic flow on Γ\*G*. We use this geometric viewpoint to establish
a correlation between measures invariant under the geodesic flow on Γ\*G*
and *T *invariant measures on the open unit interval. We use this measure
correlation to give a dynamic proof of the ergodicity of the continued fraction
map with respect to the additive Haar measure on the open unit interval. We
further discuss certain geodesics exhibiting escape of mass and derive
conclusions about continued fractions of certain algebraic elements in *L*.