Dani S.G., Nogueira Arnaldo.
On invariant measures of the Euclidean algorithm.

We study the ergodic properties of the additive Euclidean algorithm f defined in . A natural extension of f is obtained using the action of SL(2,) on a subset of SL(2,). We prove that even though f is ergodic and has an infinite invariant measure equivalent to the Lebesgue measure, such a measure is not unique; (in fact there is a continuous family of such measures). While it is folklore that this could happen for a map which is not conservative, as is the case with f, there seems to be no recorded example in the literature to that effect, and f provides a natural example for which it is the case.