Farey maps, Diophantine approximation and Bruhat–Tits tree

Based on Broise-Alamichel and Paulin's work on the Gauss map corresponding to the principal convergents via the symbolic coding of the geodesic flow of the continued fraction algorithm for formal power series with coefficients in a finite field, we continue the study of the Gauss map via Farey maps to contain all the intermediate convergents. We define the geometric Farey map, which is given by time-one map of the geodesic flow. We also define algebraic Farey maps, better suited for arithmetic properties, which produce all the intermediate convergents. Then we obtain the ergodic invariant measures for the Farey maps and the convergent speed.