Ergodic theory of the BCZ map.

In their study of Farey fracitons, Boca-Cobeli-Zaharescu introduced a piecewise-linear Lebesgue measure preserving map of the Farey triangle {x, y: 0 x, y = 1, x+y 1}. They posed the question of ergodicity of this map. In joint work with Yitwah Cheung, we show that the map is ergodic with respect to Lebesgue measure by displaying it as a first return map for horocycle flow on the modular surface. We obtain corollaries on the distribution of gaps for Farey fractions which had previously been known using analytic number theory methods.
We also obtain results on cusp excursions for horocycles.