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  2018, 12: 261-283. doi: 10.3934/jmd.2018010

Teichmüller geodesics with $ d$-dimensional limit sets

Anna Lenzhen 1, , Babak Modami 2, and  Kasra Rafi 3,

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

Received  September 04, 2016 Revised  May 07, 2018 Published  August 2018

Fund Project: BM: Partially supported by NSF grant DMS-1065872.
KR: Partially supported by NSERC Discovery grant RGPIN-435885.

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
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.  Google Scholar

[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.  Google Scholar

[3]

J Brock, C. Leininger, B. Modami and K. Rafi, Limit sets of Weil-Petersson geodesics with nonminimal ending laminations, arXiv: 1711.01663, 2017. Google Scholar

[4]

F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math., 92 (1988), 139-162.  doi: 10.1007/BF01393996.  Google Scholar

[5]

P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, 106, Birkhäuser Boston Inc., Boston, MA, 1992.  Google Scholar

[6]

J. Chaika, H. Masur and M. Wolf, Limits in $ \mathscr{PMF}$ of Teichmüller geodesics, preprint, arXiv: 1406.0564, 2014. Google Scholar

[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.  Google Scholar

[8]

A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les Surfaces, Astérisque, 66–67, Société Mathématique de France, 1979.  Google Scholar

[9]

F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Mathematical Surveys and Monographs, 76, American Mathematical Society, Providence, RI, 2000.  Google Scholar

[10]

J. Hubbard and H. A. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274.  doi: 10.1007/BF02395062.  Google Scholar

[11]

S. P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41.  doi: 10.1016/0040-9383(80)90029-4.  Google Scholar

[12]

A. Ya. Khinchin, Continued Fractions, The University of Chicago Press, Chicago, Ill.-London, 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

H. A. Masur, Two boundaries of Teichmüller space, Duke Math. J., 49 (1982), 183-190.  doi: 10.1215/S0012-7094-82-04912-2.  Google Scholar

[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.  Google Scholar

[18]

Y. N. Minsky, Harmonic maps, length, and energy in Teichmüller space, J. Differential Geom., 35 (1992), 151-217.  doi: 10.4310/jdg/1214447809.  Google Scholar

[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.  Google Scholar

[20]

K. Rafi, A combinatorial model for the Teichmüller metric, Geom. Funct. Anal., 17 (2007), 936-959.  doi: 10.1007/s00039-007-0615-x.  Google Scholar

[21]

K. Rafi, Thick-thin decomposition for quadratic differentials, Math. Res. Lett., 14 (2007), 333-341.  doi: 10.4310/MRL.2007.v14.n2.a14.  Google Scholar

[22]

K. Rafi, Hyperbolicity in Teichmüller space, Geom. Topol., 18 (2014), 3025-3053.  doi: 10.2140/gt.2014.18.3025.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

J Brock, C. Leininger, B. Modami and K. Rafi, Limit sets of Weil-Petersson geodesics with nonminimal ending laminations, arXiv: 1711.01663, 2017. Google Scholar

[4]

F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math., 92 (1988), 139-162.  doi: 10.1007/BF01393996.  Google Scholar

[5]

P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, 106, Birkhäuser Boston Inc., Boston, MA, 1992.  Google Scholar

[6]

J. Chaika, H. Masur and M. Wolf, Limits in $ \mathscr{PMF}$ of Teichmüller geodesics, preprint, arXiv: 1406.0564, 2014. Google Scholar

[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.  Google Scholar

[8]

A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les Surfaces, Astérisque, 66–67, Société Mathématique de France, 1979.  Google Scholar

[9]

F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Mathematical Surveys and Monographs, 76, American Mathematical Society, Providence, RI, 2000.  Google Scholar

[10]

J. Hubbard and H. A. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274.  doi: 10.1007/BF02395062.  Google Scholar

[11]

S. P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41.  doi: 10.1016/0040-9383(80)90029-4.  Google Scholar

[12]

A. Ya. Khinchin, Continued Fractions, The University of Chicago Press, Chicago, Ill.-London, 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

H. A. Masur, Two boundaries of Teichmüller space, Duke Math. J., 49 (1982), 183-190.  doi: 10.1215/S0012-7094-82-04912-2.  Google Scholar

[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.  Google Scholar

[18]

Y. N. Minsky, Harmonic maps, length, and energy in Teichmüller space, J. Differential Geom., 35 (1992), 151-217.  doi: 10.4310/jdg/1214447809.  Google Scholar

[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.  Google Scholar

[20]

K. Rafi, A combinatorial model for the Teichmüller metric, Geom. Funct. Anal., 17 (2007), 936-959.  doi: 10.1007/s00039-007-0615-x.  Google Scholar

[21]

K. Rafi, Thick-thin decomposition for quadratic differentials, Math. Res. Lett., 14 (2007), 333-341.  doi: 10.4310/MRL.2007.v14.n2.a14.  Google Scholar

[22]

K. Rafi, Hyperbolicity in Teichmüller space, Geom. Topol., 18 (2014), 3025-3053.  doi: 10.2140/gt.2014.18.3025.  Google Scholar

[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.  Google Scholar

Figure 1.  Case $d = 2$. The surface $X_0$ is glued out of three tori
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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
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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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