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Limits in 𝒫ℳℱ of Teichmüller geodesics

  • Jon Chaika EMAIL logo , Howard Masur and Michael Wolf
Published/Copyright: July 12, 2016

Abstract

In this paper we consider the limit set in Thurston’s compactification 𝒫 of Teichmüller space of some Teichmüller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that (a) there are quadratic differentials so that the limit set of the geodesic is a unique point, (b) there are quadratic differentials so that the limit set is a line segment, (c) there are quadratic differentials so that the vertical foliation is ergodic and there is a line segment as limit set, and (d) there are quadratic differentials so that the vertical foliation is ergodic and there is a unique point as its limit set. These give examples of divergent Teichmüller geodesics whose limit sets overlap and Teichmüller geodesics that stay a bounded distance apart but whose limit sets are not equal. A byproduct of our methods is a construction of a Teichmüller geodesic and a simple closed curve γ so that the hyperbolic length of the geodesic in the homotopy class of γ varies between increasing and decreasing on an unbounded sequence of time intervals along the geodesic.

Award Identifier / Grant number: DMS-1107452

Award Identifier / Grant number: DMS-1107263

Award Identifier / Grant number: DMS-1107367

Award Identifier / Grant number: DMS-1004372

Award Identifier / Grant number: DMS-1300550

Award Identifier / Grant number: DMS-0905907

Award Identifier / Grant number: DMS-1205016

Award Identifier / Grant number: DMS-1007383

Funding statement: Research of Jon Chaika partially supported by DMS-1004372 and DMS-1300550. Research of Howard Masur partially supported by DMS-0905907 and DMS-1205016. Research of Michael Wolf partially supported by DMS-1007383 and the Morningside Center (Tsinghua University). Howard Masur and Michael Wolf appreciate the support of the GEAR Network (DMS-1107452, DMS-1107263, DMS-1107367).

Acknowledgements

We would like to thank the referee for numerous helpful suggestions.

References

[1] L. Bers, Spaces of degenerating Riemann surfaces, Discontinuous groups and Riemann surfaces, Ann. of Math. Stud. 79, American Mathematical Society, Providence (1974), 43–55. 10.1515/9781400881642-006Search in Google Scholar

[2] D. Dumas and M. Wolf, Projective structures, grafting and measured laminations, Geom. Topol. 12 (2008), no. 1, 351–386. 10.2140/gt.2008.12.351Search in Google Scholar

[3] A. Fathi, Travaux de Thurston sur les surfaces, Astérisque 66–67, Société Mathématique de France, Paris 1979. Search in Google Scholar

[4] J. D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Math. 352, Springer, New York 1973. 10.1007/BFb0060090Search in Google Scholar

[5] F. Gardiner, Measured foliations and the minimal norm property for quadratic differentials, Acta Math. 152 (1984), 57–76. 10.1007/BF02392191Search in Google Scholar

[6] F. Gardiner, Teichmüller theory and quadratic differential, John Wiley & Sons, New York 1987. Search in Google Scholar

[7] J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), 221–274. 10.1007/BF02395062Search in Google Scholar

[8] N. V. Ivanov, Isometries of Teichmüller spaces from the point of view of Mostow rigidity, Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2 202, American Mathematical Society, Providence (2001), 131–149. 10.1090/trans2/202/11Search in Google Scholar

[9] S. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980), 23–41. 10.1016/0040-9383(80)90029-4Search in Google Scholar

[10] S. Kerckhoff, Earthquakes are analytic, Comment. Math. Helv. 60 (1985), 17–30. 10.1007/BF02567397Search in Google Scholar

[11] A. Khinchin, Continued fractions, University of Chicago Press, Chicago 1964. 10.1063/1.3051235Search in Google Scholar

[12] A. Lenzhen, Teichmüller geodesics that do not have a limit in 𝒫, Geom. Topol. 121 (2008), 177–197. 10.2140/gt.2008.12.177Search in Google Scholar

[13] A. Lenzhen, C. Leininger and K. Rafi, Limit sets of Teichmüller geodesics with minimal non-uniquely ergodic vertical foliation, J. reine angew. Math. (2015), 10.1515/crelle-2015-0040. 10.1515/crelle-2015-0040Search in Google Scholar

[14] A. Lenzhen and H. Masur, Criteria for the divergence of pairs of Teichmüller geodesics, Geom. Dedicata 144 (2010), 191–210. 10.1007/s10711-009-9397-7Search in Google Scholar

[15] A. Lenzhen and K. Rafi, Length of a curve is quasi-convex along a Teichmüller geodesic, J. Differential Geom. 88 (2011), no. 2, 267–295. 10.4310/jdg/1320067648Search in Google Scholar

[16] B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381–386. 10.5186/aasfm.1985.1042Search in Google Scholar

[17] H. Masur, The extension of the Weil–Petersson metric to the boundary of Teichmüller space, Duke Math. J. 43 (1976), 623–635. 10.1215/S0012-7094-76-04350-7Search in Google Scholar

[18] H. Masur, Two boundaries of Teichmüller space, Duke Math. J. 49 (1982), no. 1, 183–190. 10.1215/S0012-7094-82-04912-2Search in Google Scholar

[19] H. Masur and J. Smillie, Hausdorff dimension of sets of foliations, Ann. of Math. (2) 134 (1991), 455–543. 10.2307/2944356Search in Google Scholar

[20] H. Masur and S. Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems. Vol. 1A, North-Holland, Amsterdam (2002), 1015–1089. 10.1016/S1874-575X(02)80015-7Search in Google Scholar

[21] K. Rafi, A characterization of short curves of a Teichmüller geodesic, Geom. Topol. 9 (2005), 179–202. 10.2140/gt.2005.9.179Search in Google Scholar

[22] K. Rafi, A Combinatorial model for Teichmüller space, Geom. Funct. Anal. 17 (2007), no. 3, 936–959. 10.1007/s00039-007-0615-xSearch in Google Scholar

[23] M. Rees, An alternative approach to the ergodic theory of measured foliations on surfaces, Ergodic Theory Dynam. Systems 1 (1981), 461–488. 10.1017/S0143385700001383Search in Google Scholar

[24] W. A. Veech, Strict ergodicity in zero dimensional dynamical systems and the Kronecker–Weyl theorem mod 2, Trans. Amer. Math. Soc. 140 (1969), 1–33. 10.1090/S0002-9947-1969-0240056-XSearch in Google Scholar

[25] W. A. Veech, Interval exchange transformations, J. Anal. Math. 33 (1978), 222–272. 10.1007/BF02790174Search in Google Scholar

[26] R. A. Wentworth, Energy of harmonic maps and Gardiner’s formula, In the tradition of Ahlfors–Bers. IV, Contemp. Math. 432, American Mathematical Society, Providence (2007), 221–229. 10.1090/conm/432/08311Search in Google Scholar

[27] M. Wolf, Infinite energy harmonic maps and degeneration of hyperbolic surfaces in moduli space, J. Differential Geom. 33 (1991), 487–539. 10.4310/jdg/1214446328Search in Google Scholar

[28] M. Wolf, The Weil–Petersson Hessian of length on Teichmüller space, J. Differential Geom. 91 (2012), no. 1, 129–169. 10.4310/jdg/1343133703Search in Google Scholar

[29] S. Yamada, Weil–Peterson convexity of the energy functional on classical and universal Teichmüller spaces, J. Differential Geom. 51 (1999), no. 1, 35–96. 10.4310/jdg/1214425025Search in Google Scholar

Received: 2014-06-17
Revised: 2015-10-19
Published Online: 2016-07-12
Published in Print: 2019-02-01

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