On invariant measures of the Euclidean algorithm
نویسندگان
چکیده
We study the ergodic properties of the additive Euclidean algorithm f defined in R+. A natural extension of f is obtained using the action of SL(2,Z) on a subset of SL(2,R). 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.
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