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In his paper "Geodesic laminations on surfaces", Bonahon gave the definition of generic arc and a property as following.

An arc $k$ is generic (with respect to simple geodesics) if it is transverse to every simple geodesic of $S$. And almost every geodesic arc is generic since the union of all simple geodesics has Hausdorff dimension 1. I am not sure why this true and what is the measure used here for geodesic arcs?

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  • $\begingroup$ What exactly is the statement: "is generic" or "has measure 1"? $\endgroup$
    – Alex Degtyarev
    Commented Aug 25, 2015 at 9:56
  • $\begingroup$ Alex, I am sorry that there is a typo in the question. The "Hausdorff measure" should be "Hasudorff dimesion". Now the question is corrected. $\endgroup$
    – user78500
    Commented Aug 25, 2015 at 10:38

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The fact that the union of all simple geodesics has Hausdorff dimension 1 is a classic result of Joan Birman and Caroline Series

Birman, Joan S., and Caroline Series. "Geodesics with bounded intersection number on surfaces are sparsely distributed." Topology 24.2 (1985): 217-225.

The measure used for geodesic arcs probably does not matter too much, but picking two points uniformly at random on the surface and connecting them with the shortest arc works fine (this only gives you length-minimizing arcs, but I don't think there is a good measure on the set of ALL geodesic arcs).

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  • $\begingroup$ Prof. Rivin, Thank you very much! May I ask you one more question? Why the Hausdorff dimension 1 implies that almost every geodesic arc is generic? Is it obvious? It seems that a generic geodesic should be very long otherwise it will not intersect with all simple geodesic. Thank you very much! $\endgroup$
    – user78500
    Commented Aug 26, 2015 at 2:09

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