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Corrected: I didn't understand that dense is not the same as probability 1.
Joseph O'Rourke
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Surfaces with many (but not solely) closed geodesics?

Let $S$ be a closed surface embedded in $\mathbb{R}^3$, let's say of genus zero. I seek examples of $S$ with the following property: If one selects a random any point $p$ on $S$, and a random direction $u$ tangent to $S$ at $p$, then the geodesic issuing from $p$ in direction $u$ is a closed geodesic with positive probability. [Question modified to reflect @alvarezpaiva's comment.]

I know this is trivially true for Zoll surfaces, on which every geodesic is closed; see figure below. But are there non-Zoll $S$ where geodesics are prevalent enough to yield a positive probability (perhaps $1$)? Or, to be more explicit:

Q. If, for every $p \in S$, geodesics issuing from $p$ occur with probablility $1$ among all tangent directions $u$, does that imply that every such geodesic is closed?


           ![ZollDiscr][1]
           (Zoll Surface: Image from [Polthier&Schmies][2] via [this MO question][3])
Joseph O'Rourke
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