Invariant Measures for the Action of SL(2,Z) on $\r^2$

A. Nogueira

1998, v.4, №4, 571-592


In [Invariant measures and minimal sets of horospherical flows, Invent. Math., 1979, 64, 353-385] S.G. Dani obtained a classification of the Borel measures, locally finite, that are ergodic for the natural action of $SL(n,Z)$ on $R^n$. This result was derived from the study of a rich class of dynamical systems: the unipotent flows acting on homogeneous spaces. Here we shall look at the action of $SL(2,Z)$, restricted to $R^{2}_{+}$. This is closely related to the horocycle flow. The ergodicity of the horocycle flow was proved by Hedlund [A metrically transitive group defined by the modular group, Amer. J. Math., 1935, 57, 668-678], and the unique ergodicity much later by Furstenberg [The unique ergodicity of the horocycle flow. In: Recent Advances in Topological Dynamics, A. Beck (ed.), Springer, Berlin, Heidelberg, New York, 95-115] Here we give an alternative approach to Dani's theorem. We look at the Borel measures, that are invariant under the action of $SL(2,Z)$. Up to a multiplicative constant, the Lebesgue measure is the only such measure, locally finite at a point, that annihilates the set of commensurable points. Regarding those, with support in the set of commensurable points, the ergodic ones are the counting measures in the orbit of a point in the diagonal, not necessarily locally finite.

Keywords: action of $SL$(2,Z),invariant measures,ergodic measures,density of relatively prime numbers


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