On the equivalence relations of αα-continued fractions

We compare the equivalence relations on real numbers arising from having eventually agreeing αα-continued fraction expansion (for each 0≤α≤10≤α≤1) with those from sharing GL(2,Z)- or SL(2,Z)-orbits. We show that the αα-relation and the GL(2,Z)-relation of xx are identical for any α>0α>0 when xx is of unbounded type. On the other hand, they are not identical for xx of bounded type.