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Dmitri Panov
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This is more of a comment intended to provide examples. One can construct a reasonably large class of exampleexamples of metrics on $T^2$ satisfying your property. I believe moreover that this class of metrics (described below) is stable with respect to small $C^{\infty}$ perturbation.

An infiniteInfinite geodesics on surfaces without self intersections aare closely connected to geodesic lamintations. Now, hyperbolic surfaces have plenty of geodesic laminations. So we can do the following. First take a hyperbolic metric on $T^2$ with a cusp and then smoothen the cusp close to infinity. Then there still will be a geodesic lamination in the hyperbolic part of $T^2$. For a concrete example, look at Fig 11 here: (Geodesic lamination on surfaces by Bonahon)

https://www-bcf.usc.edu/~fbonahon/Research/Preprints/StonyBrookProc.ps


          [![Fig11][1]][1]
          Fig.11. A geodesic lamination on the punctured torus.

Or one can use a different construction, identifying a part of $T^2$ with a part of hyperbolic surface of genus $2$ as on Fig 5 in the same text. In this case one will get in infinite geodesic in $T^2$ that accumulates to two closed geodesics on both ends.


          [![Fig5][2]][2]

I believe that both examples are stable under small perturbation of the metric.

This is more of a comment intended to provide examples. One can construct a reasonably large class of example of metrics on $T^2$ satisfying your property. I believe moreover that this class of metrics is stable with respect to small $C^{\infty}$ perturbation.

An infinite geodesics on surfaces without self intersections a closely connected to geodesic lamintations. Now, hyperbolic surfaces have plenty of geodesic laminations. So we can do the following. First take a hyperbolic metric on $T^2$ with a cusp and then smoothen the cusp close to infinity. Then there still will be a geodesic lamination in the hyperbolic part of $T^2$. For a concrete example, look at Fig 11 here: (Geodesic lamination on surfaces by Bonahon)

https://www-bcf.usc.edu/~fbonahon/Research/Preprints/StonyBrookProc.ps


          [![Fig11][1]][1]
          Fig.11. A geodesic lamination on the punctured torus.

Or one can use a different construction, identifying a part of $T^2$ with a part of hyperbolic surface of genus $2$ as on Fig 5 in the same text. In this case one will get in infinite geodesic in $T^2$ that accumulates to two closed geodesics on both ends.


          [![Fig5][2]][2]

I believe that both examples are stable under small perturbation of the metric.

This is more of a comment intended to provide examples. One can construct a reasonably large class of examples of metrics on $T^2$ satisfying your property. I believe moreover that this class of metrics (described below) is stable with respect to small $C^{\infty}$ perturbation.

Infinite geodesics on surfaces without self intersections are closely connected to geodesic lamintations. Now, hyperbolic surfaces have plenty of geodesic laminations. So we can do the following. First take a hyperbolic metric on $T^2$ with a cusp and then smoothen the cusp close to infinity. Then there still will be a geodesic lamination in the hyperbolic part of $T^2$. For a concrete example, look at Fig 11 here: (Geodesic lamination on surfaces by Bonahon)

https://www-bcf.usc.edu/~fbonahon/Research/Preprints/StonyBrookProc.ps


          [![Fig11][1]][1]
          Fig.11. A geodesic lamination on the punctured torus.

Or one can use a different construction, identifying a part of $T^2$ with a part of hyperbolic surface of genus $2$ as on Fig 5 in the same text. In this case one will get in infinite geodesic in $T^2$ that accumulates to two closed geodesics on both ends.


          [![Fig5][2]][2]

I believe that both examples are stable under small perturbation of the metric.

Added the cited Fig.11 & Fig.5.
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Joseph O'Rourke
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This is more of a comment intended to provide examples. One can construct a reasonably large class of example of metrics on $T^2$ satisfying your property. I believe moreover that this class of metrics is stable with respect to small $C^{\infty}$ perturbation.

An infinite geodesics on surfaces without self intersections a closely connected to geodesic lamintations. Now, hyperbolic surfaces have plenty of geodesic laminations. So we can do the following. First take a hyperbolic metric on $T^2$ with a cusp and then smoothen the cusp close to infinity. Then there still will be a geodesic lamination in the hyperbolic part of $T^2$. For a concrete example, look at Fig 11 here: (Geodesic lamination on surfaces by Bonahon)

https://www-bcf.usc.edu/~fbonahon/Research/Preprints/StonyBrookProc.ps


          [![Fig11][1]][1]
          Fig.11. A geodesic lamination on the punctured torus.

Or one can use a different construction, identifying a part of $T^2$ with a part of hyperbolic surface of genus $2$ as on Fig 5 in the same text. In this case one will get in infinite geodesic in $T^2$ that accumulates to two closed geodesics on both ends.


          [![Fig5][2]][2]

I believe that both examples are stable under small perturbation of the metric.

This is more of a comment intended to provide examples. One can construct a reasonably large class of example of metrics on $T^2$ satisfying your property. I believe moreover that this class of metrics is stable with respect to small $C^{\infty}$ perturbation.

An infinite geodesics on surfaces without self intersections a closely connected to geodesic lamintations. Now, hyperbolic surfaces have plenty of geodesic laminations. So we can do the following. First take a hyperbolic metric on $T^2$ with a cusp and then smoothen the cusp close to infinity. Then there still will be a geodesic lamination in the hyperbolic part of $T^2$. For a concrete example, look at Fig 11 here: (Geodesic lamination on surfaces by Bonahon)

https://www-bcf.usc.edu/~fbonahon/Research/Preprints/StonyBrookProc.ps

Or one can use a different construction, identifying a part of $T^2$ with a part of hyperbolic surface of genus $2$ as on Fig 5 in the same text. In this case one will get in infinite geodesic in $T^2$ that accumulates to two closed geodesics on both ends.

I believe that both examples are stable under small perturbation of the metric.

This is more of a comment intended to provide examples. One can construct a reasonably large class of example of metrics on $T^2$ satisfying your property. I believe moreover that this class of metrics is stable with respect to small $C^{\infty}$ perturbation.

An infinite geodesics on surfaces without self intersections a closely connected to geodesic lamintations. Now, hyperbolic surfaces have plenty of geodesic laminations. So we can do the following. First take a hyperbolic metric on $T^2$ with a cusp and then smoothen the cusp close to infinity. Then there still will be a geodesic lamination in the hyperbolic part of $T^2$. For a concrete example, look at Fig 11 here: (Geodesic lamination on surfaces by Bonahon)

https://www-bcf.usc.edu/~fbonahon/Research/Preprints/StonyBrookProc.ps


          [![Fig11][1]][1]
          Fig.11. A geodesic lamination on the punctured torus.

Or one can use a different construction, identifying a part of $T^2$ with a part of hyperbolic surface of genus $2$ as on Fig 5 in the same text. In this case one will get in infinite geodesic in $T^2$ that accumulates to two closed geodesics on both ends.


          [![Fig5][2]][2]

I believe that both examples are stable under small perturbation of the metric.

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Dmitri Panov
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This is more of a comment intended to provide examples. One can construct a reasonably large class of example of metrics on $T^2$ satisfying your property. I believe moreover that this class of metrics is stable with respect to small $C^{\infty}$ perturbation.

An infinite geodesics on surfaces without self intersections a closely connected to geodesic lamintations. Now, hyperbolic surfaces have plenty of geodesic laminations. So we can do the following. First take a hyperbolic metric on $T^2$ with a cusp and then smoothen the cusp close to infinity. Then there still will be a geodesic lamination in the hyperbolic part of $T^2$. For a concrete example, look at Fig 11 here: (Geodesic lamination on surfaces by Bonahon)

https://www-bcf.usc.edu/~fbonahon/Research/Preprints/StonyBrookProc.ps

Or one can use a different construction, identifying a part of $T^2$ with a part of hyperbolic surface of genus $2$ as on Fig 5 in the same text. In this case one will get in infinite geodesic in $T^2$ that accumulates to two closed geodesics on both ends.

I believe that both examples are stable under small perturbation of the metric.