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2018, 12: 261-283.
doi: 10.3934/jmd.2018010
Teichmüller geodesics with $ d$-dimensional limit sets
| 1. | Laboratoire de Mathématiques, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France |
| 2. | Mathematics Department, Stony Brook, University, Stony Brook, NY 11794-3651, USA |
| 3. | Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada |
We construct an example of a Teichmüller geodesic ray whose limit set in the Thurston boundary of Teichmüller space is a $ d$-dimensional simplex.
Keywords:
Teichmüller geodesics,
Thurston boundary of Teichmüller space,
nonuniquely ergodic measured foliations,
quadratic differentials.
Mathematics Subject Classification:
Primary: 32G15; Secondary: 57M50.
Citation:
Anna Lenzhen, Babak Modami, Kasra Rafi. Teichmüller geodesics with $ d$-dimensional limit sets. Journal of Modern Dynamics,
2018, 12: 261-283.
doi: 10.3934/jmd.2018010
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References:
| [1] |
J. Brock, C. Leininger, B. Modami and K. Rafi,
Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, Ⅱ, J. Reine. Angew. Math., 737 (2018), 1-32.
doi: 10.1515/crelle-2015-0040. |
| [2] |
J. Brock, C. Leininger, B. Modami and K. Rafi, Limit sets of Weil-Petersson geodesics, Int. Math. Res. Not. IMRN, (2018), arXiv: 1611.02197.
doi: 10.1093/imrn/rny002. |
| [3] |
J Brock, C. Leininger, B. Modami and K. Rafi, Limit sets of Weil-Petersson geodesics with nonminimal ending laminations, arXiv: 1711.01663, 2017. |
| [4] |
F. Bonahon,
The geometry of Teichmüller space via geodesic currents, Invent. Math., 92 (1988), 139-162.
doi: 10.1007/BF01393996. |
| [5] |
P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, 106, Birkhäuser Boston Inc., Boston, MA, 1992. |
| [6] |
J. Chaika, H. Masur and M. Wolf, Limits in $ \mathscr{PMF}$ of Teichmüller geodesics, preprint, arXiv: 1406.0564, 2014. |
| [7] |
Y.-E. Choi, K. Rafi and C. Series,
Lines of minima and Teichmüller geodesics, Geom. Funct. Anal., 18 (2008), 698-754.
doi: 10.1007/s00039-008-0675-6. |
| [8] |
A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les Surfaces, Astérisque, 66–67, Société Mathématique de France, 1979. |
| [9] |
F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Mathematical Surveys and Monographs, 76, American Mathematical Society, Providence, RI, 2000. |
| [10] |
J. Hubbard and H. A. Masur,
Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274.
doi: 10.1007/BF02395062. |
| [11] |
S. P. Kerckhoff,
The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41.
doi: 10.1016/0040-9383(80)90029-4. |
| [12] |
A. Ya. Khinchin, Continued Fractions, The University of Chicago Press, Chicago, Ill.-London, 1964. |
| [13] |
A. Lenzhen,
Teichmüller geodesics that do not have a limit in $ \mathscr{PMF}$, Geom. Topol., 12 (2008), 177-197.
doi: 10.2140/gt.2008.12.177. |
| [14] |
C. Leininger, A. Lenzhen and K. Rafi,
Limit sets of Teichmüller geodesics with minimal non-uniquely ergodic vertical foliation, J. Reine. Angew. Math., 737 (2018), 1-32.
doi: 10.1515/crelle-2015-0040. |
| [15] |
A. Lenzhen, K. Rafi and J. Tao,
Bounded combinatorics and the Lipschitz metric on Teichmüller space, Geom. Dedicata, 159 (2012), 353-371.
doi: 10.1007/s10711-011-9664-2. |
| [16] |
H. A. Masur,
Two boundaries of Teichmüller space, Duke Math. J., 49 (1982), 183-190.
doi: 10.1215/S0012-7094-82-04912-2. |
| [17] |
B. Maskit,
Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 10 (1985), 381-386.
doi: 10.5186/aasfm.1985.1042. |
| [18] |
Y. N. Minsky,
Harmonic maps, length, and energy in Teichmüller space, J. Differential Geom., 35 (1992), 151-217.
doi: 10.4310/jdg/1214447809. |
| [19] |
H. A. Masur and Y. N. Minsky,
Geometry of the complex of curves. I. Hyperbolicity, Invent. Math., 138 (1999), 103-149.
doi: 10.1007/s002220050343. |
| [20] |
K. Rafi,
A combinatorial model for the Teichmüller metric, Geom. Funct. Anal., 17 (2007), 936-959.
doi: 10.1007/s00039-007-0615-x. |
| [21] |
K. Rafi,
Thick-thin decomposition for quadratic differentials, Math. Res. Lett., 14 (2007), 333-341.
doi: 10.4310/MRL.2007.v14.n2.a14. |
| [22] |
K. Rafi,
Hyperbolicity in Teichmüller space, Geom. Topol., 18 (2014), 3025-3053.
doi: 10.2140/gt.2014.18.3025. |
| [23] |
S. A. Wolpert,
The length spectra as moduli for compact Riemann surfaces, Ann. of Math. (2), 109 (1979), 323-351.
doi: 10.2307/1971114. |
show all references
References:
| [1] |
J. Brock, C. Leininger, B. Modami and K. Rafi,
Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, Ⅱ, J. Reine. Angew. Math., 737 (2018), 1-32.
doi: 10.1515/crelle-2015-0040. |
| [2] |
J. Brock, C. Leininger, B. Modami and K. Rafi, Limit sets of Weil-Petersson geodesics, Int. Math. Res. Not. IMRN, (2018), arXiv: 1611.02197.
doi: 10.1093/imrn/rny002. |
| [3] |
J Brock, C. Leininger, B. Modami and K. Rafi, Limit sets of Weil-Petersson geodesics with nonminimal ending laminations, arXiv: 1711.01663, 2017. |
| [4] |
F. Bonahon,
The geometry of Teichmüller space via geodesic currents, Invent. Math., 92 (1988), 139-162.
doi: 10.1007/BF01393996. |
| [5] |
P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, 106, Birkhäuser Boston Inc., Boston, MA, 1992. |
| [6] |
J. Chaika, H. Masur and M. Wolf, Limits in $ \mathscr{PMF}$ of Teichmüller geodesics, preprint, arXiv: 1406.0564, 2014. |
| [7] |
Y.-E. Choi, K. Rafi and C. Series,
Lines of minima and Teichmüller geodesics, Geom. Funct. Anal., 18 (2008), 698-754.
doi: 10.1007/s00039-008-0675-6. |
| [8] |
A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les Surfaces, Astérisque, 66–67, Société Mathématique de France, 1979. |
| [9] |
F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Mathematical Surveys and Monographs, 76, American Mathematical Society, Providence, RI, 2000. |
| [10] |
J. Hubbard and H. A. Masur,
Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274.
doi: 10.1007/BF02395062. |
| [11] |
S. P. Kerckhoff,
The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41.
doi: 10.1016/0040-9383(80)90029-4. |
| [12] |
A. Ya. Khinchin, Continued Fractions, The University of Chicago Press, Chicago, Ill.-London, 1964. |
| [13] |
A. Lenzhen,
Teichmüller geodesics that do not have a limit in $ \mathscr{PMF}$, Geom. Topol., 12 (2008), 177-197.
doi: 10.2140/gt.2008.12.177. |
| [14] |
C. Leininger, A. Lenzhen and K. Rafi,
Limit sets of Teichmüller geodesics with minimal non-uniquely ergodic vertical foliation, J. Reine. Angew. Math., 737 (2018), 1-32.
doi: 10.1515/crelle-2015-0040. |
| [15] |
A. Lenzhen, K. Rafi and J. Tao,
Bounded combinatorics and the Lipschitz metric on Teichmüller space, Geom. Dedicata, 159 (2012), 353-371.
doi: 10.1007/s10711-011-9664-2. |
| [16] |
H. A. Masur,
Two boundaries of Teichmüller space, Duke Math. J., 49 (1982), 183-190.
doi: 10.1215/S0012-7094-82-04912-2. |
| [17] |
B. Maskit,
Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 10 (1985), 381-386.
doi: 10.5186/aasfm.1985.1042. |
| [18] |
Y. N. Minsky,
Harmonic maps, length, and energy in Teichmüller space, J. Differential Geom., 35 (1992), 151-217.
doi: 10.4310/jdg/1214447809. |
| [19] |
H. A. Masur and Y. N. Minsky,
Geometry of the complex of curves. I. Hyperbolicity, Invent. Math., 138 (1999), 103-149.
doi: 10.1007/s002220050343. |
| [20] |
K. Rafi,
A combinatorial model for the Teichmüller metric, Geom. Funct. Anal., 17 (2007), 936-959.
doi: 10.1007/s00039-007-0615-x. |
| [21] |
K. Rafi,
Thick-thin decomposition for quadratic differentials, Math. Res. Lett., 14 (2007), 333-341.
doi: 10.4310/MRL.2007.v14.n2.a14. |
| [22] |
K. Rafi,
Hyperbolicity in Teichmüller space, Geom. Topol., 18 (2014), 3025-3053.
doi: 10.2140/gt.2014.18.3025. |
| [23] |
S. A. Wolpert,
The length spectra as moduli for compact Riemann surfaces, Ann. of Math. (2), 109 (1979), 323-351.
doi: 10.2307/1971114. |
Figure 2.
Annuli in $T^i$ about $\beta^i$ and $\alpha_n^i$ . Expanding annulus with core curve $\beta^i$ , on the left, and flat annulus about $\alpha_n^i$ , on the right, at the time when $\alpha_n^i$ is balanced
Figure 3.
The interval when $\alpha_k^i$ is short. For $k = 2n$ , the curves $\alpha_k^i,\; i\in\mathbb{Z}_3$ , start getting short at different times, but they grow back to length 1 roughly at the same time. We choose $t_n$ in the shaded interval to guarantee that all three curves are short and have collar neighborhoods of approximately the same width
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