Skip to main content
×
×
Home

Indiscriminate covers of infinite translation surfaces are innocent, not devious

  • W. PATRICK HOOPER (a1) (a2) and RODRIGO TREVIÑO (a3)
Abstract

We consider the interaction between passing to finite covers and ergodic properties of the straight-line flow on finite-area translation surfaces with infinite topological type. Infinite type provides for a rich family of degree- $d$ covers for any integer $d>1$ . We give examples which demonstrate that passing to a finite cover can destroy ergodicity, but we also provide evidence that this phenomenon is rare. We define a natural notion of a random degree $d$ cover and show that, in many cases, ergodicity and unique ergodicity are preserved under passing to random covers. This work provides a new context for exploring the relationship between recurrence of the Teichmüller flow and ergodic properties of the straight-line flow.

Copyright
References
Hide All
[BM74] Beardon, A. F. and Maskit, B.. Limit points of Kleinian groups and finite sided fundamental polyhedra. Acta Math. 132 (1974), 112.
[Bow13] Bowman, J. P.. The complete family of Arnoux–Yoccoz surfaces. Geom. Dedicata 164(1) (2013), 113130.
[Buf13] Bufetov, A. I.. Limit theorems for suspension flows over Vershik automorphisms. Russian Math. Surveys 68(5) (2013), 789860.
[CE07] Cheung, Y. and Eskin, A.. Unique ergodicity of translation flows. Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow. Selected Papers of the Workshop, Toronto, Ontario, Canada, January 2006 (Fields Institute Communications, 51) . Ed. Giovanni, F. et al. . American Mathematical Society, Providence, RI, 2007, pp. 213221.
[CGL06] Chamanara, R., Gardiner, F. P. and Lakic, N.. A hyperelliptic realization of the horseshoe and baker maps. Ergod. Th. & Dynam. Sys. 26(6) (2006), 17491768.
[Cha04] Chamanara, R.. Affine automorphism groups of surfaces of infinite type. In the Tradition of Ahlfors and Bers, III (Contemporary Mathematics, 355) . American Mathematical Society, Providence, RI, 2004, pp. 123145.
[DEDML98] Degli Esposti, M., Del Magno, G. and Lenci, M.. An infinite step billiard. Nonlinearity 11(4) (1998), 9911013.
[HR16] Herrlich, F. and Randecker, A.. Notes on the veech group of the chamanara surface, Preprint, 2016, arXiv:1612.06877.
[Hoo13] Hooper, W. P.. Immersions and the space of all translation structures, Preprint, 2013, arXiv:1310.5193.
[Hoo15] Hooper, W. P.. The invariant measures of some infinite interval exchange maps. Geom. Topol. 19(4) (2015), 18952038.
[Hoo16] Hooper, W. Patrick. Immersions and translation structures I: The space of structures on the pointed disk, Preprint, 2016, arXiv:1309.4795.
[Hub06] Hubbard, J. H.. Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Vol. 1: Teichmüller theory. Matrix Editions, Ithaca, NY, 2006.
[HW13] Hubert, P. and Weiss, B.. Ergodicity for infinite periodic translation surfaces. Compos. Math. 149(8) (2013), 13641380 (in English).
[LTn16] Lindsey, K. and Treviño, R.. Infinite type flat surface models of ergodic systems. Discrete Contin. Dyn. Syst. 36(10) (2016), 55095553.
[Mas92] Masur, H.. Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. Duke Math. J. 66(3) (1992), 387442.
[MS91] Masur, H. and Smillie, J.. Hausdorff dimension of sets of nonergodic measured foliations. Ann. of Math. (2) 134(3) (1991), 455543.
[MT02] Masur, H. and Tabachnikov, S.. Rational billiards and flat structures. Handbook of Dynamical Systems, Vol. 1A. North-Holland, Amsterdam, 2002, pp. 10151089.
[PSV11] Przytycki, P., Schmithüsen, G. and Valdez, F.. Veech groups of Loch Ness monsters. Ann. Inst. Fourier 61(2) (2011), 673687 (in English).
[Pud13] Puder, D.. Primitive words, free factors and measure preservation. Israel J. Math. (2013), 149 (in English).
[Rad25] Radó, T.. Über den Begriff der Riemannschen Fläche. Acta Litt. Sci. Szeged 2 (1925), 101121, 10.
[Ran14] Randecker, A.. Wild translation surfaces and infinite genus, Preprint, 2014, arXiv:1410.1501.
[Ric63] Richards, I.. On the classification of noncompact surfaces. Trans. Amer. Math. Soc. 106(2) (1963), 259269.
[Rya97] Ryan, K.. Elephant Rocks. Grove Press, New York, 1997.
[Thu88] Thurston, W. P.. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19(2) (1988), 417431.
[Tre14] Treviño, R.. On the ergodicity of flat surfaces of finite area. Geom. Funct. Anal. (2014), 127.
[Tre16] Treviño, R.. Flat surfaces, Bratteli diagrams, and unique ergodicity à la Masur. Israel J. Math. (2017), to appear, arXiv:1604.03572.
[Tro99] Troubetzkoy, S.. Billiards in infinite polygons. Nonlinearity 12(3) (1999), 513524.
[Vee89] Veech, W. A.. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97(3) (1989), 553583.
[VJ68] Vere-Jones, D.. Ergodic properties of nonnegative matrices. II. Pacific J. Math. 26 (1968), 601620.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 8 *
Loading metrics...

Abstract views

Total abstract views: 91 *
Loading metrics...

* Views captured on Cambridge Core between 4th December 2017 - 22nd July 2018. This data will be updated every 24 hours.