July 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

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.

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. BrockC. LeiningerB. 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. ChoiK. 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. LeiningerA. 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. LenzhenK. 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. BrockC. LeiningerB. 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. ChoiK. 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. LeiningerA. 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. LenzhenK. 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 1.  Case $d = 2$. The surface $X_0$ is glued out of three tori
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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