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Based on Broise-Alamichel and Paulin's work on the Gauss map corresponding tothe principal convergents, we continue the study of the Gauss map via Fareymaps to contain all the intermediate convergents. We define the geometric Fareymap, which is given by time-1 map of the geodesic flow. We also definealgebraic Farey maps, better suited for arithmetic properties, which produceall the intermediate convergents. Then we obtain the ergodic invariant measuresfor the Farey maps and the convergent speed.

Farey map, Diophantine approximation and Bruhat-Tits tree