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The invariant measures of some infinite interval exchange maps. (English) Zbl 1371.37076

Summary: We classify the locally finite ergodic invariant measures of certain infinite interval exchange transformations (IETs). These transformations naturally arise from return maps of the straight-line flow on certain translation surfaces, and the study of the invariant measures for these IETs is equivalent to the study of invariant measures for the straight-line flow in some direction on these translation surfaces. For the surfaces and directions to which our methods apply, we can characterize the locally finite ergodic invariant measures of the straight-line flow in a set of directions of Hausdorff dimension larger than \(\frac{1}{2}\). We promote this characterization to a classification in some cases. For instance, when the surfaces admit a cocompact action by a nilpotent group, we prove each ergodic invariant measure for the straight-line flow is a Maharam measure, and we describe precisely which Maharam measures arise. When the surfaces under consideration are of finite area, the straight-line flows in the directions we understand are uniquely ergodic. Our methods apply to translation surfaces admitting multitwists in a pair of cylinder decompositions in nonparallel directions.

MSC:

37E05 Dynamical systems involving maps of the interval
37E20 Universality and renormalization of dynamical systems
37A40 Nonsingular (and infinite-measure preserving) transformations
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[1] J Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs 50, Amer. Math. Soc. (1997) · Zbl 0882.28013 · doi:10.1090/surv/050
[2] J Aaronson, H Nakada, O Sarig, R Solomyak, Invariant measures and asymptotics for some skew products, Israel J. Math. 128 (2002) 93 · Zbl 1006.28013 · doi:10.1007/BF02785420
[3] C D Aliprantis, R Tourky, Cones and duality, Graduate Studies in Mathematics 84, Amer. Math. Soc. (2007) · Zbl 1127.46002 · doi:10.1090/gsm/084
[4] A F Beardon, B Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974) 1 · Zbl 0277.30017 · doi:10.1007/BF02392106
[5] F Bonahon, Geodesic laminations on surfaces (editors M Lyubich, J W Milnor, Y N Minsky), Contemp. Math. 269, Amer. Math. Soc. (2001) 1 · Zbl 0996.53029 · doi:10.1090/conm/269/04327
[6] J P Bowman, The complete family of Arnoux-Yoccoz surfaces, Geom. Dedicata 164 (2013) 113 · Zbl 1277.30026 · doi:10.1007/s10711-012-9762-9
[7] J P Bowman, F Valdez, Wild singularities of flat surfaces, Israel J. Math. 197 (2013) 69 · Zbl 1284.30039 · doi:10.1007/s11856-013-0022-y
[8] D Burago, Y Burago, S Ivanov, A course in metric geometry, Graduate Studies in Mathematics 33, Amer. Math. Soc. (2001) · Zbl 0981.51016 · doi:10.1090/gsm/033
[9] R Chamanara, Affine automorphism groups of surfaces of infinite type (editors W Abikoff, A Haas), Contemp. Math. 355, Amer. Math. Soc. (2004) 123 · Zbl 1069.30073 · doi:10.1090/conm/355/06449
[10] R Chamanara, F P Gardiner, N Lakic, A hyperelliptic realization of the horseshoe and baker maps, Ergodic Theory Dynam. Systems 26 (2006) 1749 · Zbl 1121.37036 · doi:10.1017/S0143385706000484
[11] Y Cheung, A Eskin, Unique ergodicity of translation flowsuller flow” (editors G Forni, M Lyubich, C Pugh, M Shub), Fields Inst. Commun. 51, Amer. Math. Soc. (2007) 213 · Zbl 1145.37003
[12] J P Conze, Recurrence, ergodicity and invariant measures for cocycles over a rotation (editor I Assani), Contemp. Math. 485, Amer. Math. Soc. (2009) 45 · Zbl 1183.37005 · doi:10.1090/conm/485/09492
[13] J P Conze, K Fr\caczek, Cocycles over interval exchange transformations and multivalued Hamiltonian flows, Adv. Math. 226 (2011) 4373 · Zbl 1236.37006 · doi:10.1016/j.aim.2010.11.014
[14] J P Conze, E Gutkin, On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces, Ergodic Theory Dynam. Systems 32 (2012) 491 · Zbl 1261.37018 · doi:10.1017/S0143385711001003
[15] J P Conze, M Keane, Ergodicité d’un flot cylindrique, Dépt. Math. Informat., Univ. Rennes (1976)
[16] B D Craven, J J Koliha, Generalizations of Farkas’ theorem, SIAM J. Math. Anal. 8 (1977) 983 · Zbl 0408.52006 · doi:10.1137/0508076
[17] V Delecroix, P Hubert, S Lelièvre, Diffusion for the periodic wind-tree model, Ann. Sci. Éc. Norm. Supér. 47 (2014) 1085 · Zbl 1351.37159
[18] J Farkas, Über die Theorie der Einfachen Ungleichungen, J. Reine Angew. Math. 124 (1902) 1 · JFM 32.0169.02
[19] K Fraczek, C Ulcigrai, Non-ergodic \(\mathbbZ\)-periodic billiards and infinite translation surfaces, Invent. Math. 197 (2014) 241 · Zbl 1316.37006 · doi:10.1007/s00222-013-0482-z
[20] J Hardy, J Weber, Diffusion in a periodic wind-tree model, J. Math. Phys. 21 (1980) 1802 · doi:10.1063/1.524633
[21] W P Hooper, Dynamics on an infinite surface with the lattice property, · Zbl 1290.30053
[22] W P Hooper, Grid graphs and lattice surfaces, Int. Math. Res. Not. 2013 (2013) 2657 · Zbl 1333.37047 · doi:10.1093/imrn/rns124
[23] W P Hooper, An infinite surface with the lattice property I: Veech groups and coding geodesics, Trans. Amer. Math. Soc. 366 (2014) 2625 · Zbl 1290.30053 · doi:10.1090/S0002-9947-2013-06139-9
[24] W P Hooper, P Hubert, B Weiss, Dynamics on the infinite staircase, Discrete Contin. Dyn. Syst. 33 (2013) 4341 · Zbl 1306.37043 · doi:10.3934/dcds.2013.33.4341
[25] W P Hooper, B Weiss, Generalized staircases: Recurrence and symmetry, Ann. Inst. Fourier (Grenoble) 62 (2012) 1581 · Zbl 1279.37035 · doi:10.5802/aif.2730
[26] P Hubert, S Lelièvre, S Troubetzkoy, The Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion, J. Reine Angew. Math. 656 (2011) 223 · Zbl 1233.37025 · doi:10.1515/CRELLE.2011.052
[27] P Hubert, G Schmithüsen, Infinite translation surfaces with infinitely generated Veech groups, J. Mod. Dyn. 4 (2010) 715 · Zbl 1219.30019 · doi:10.3934/jmd.2010.4.715
[28] A Katok, B Hasselblatt, Introduction to the modern theory of dynamical systems, Encycl. Math. and Appl. 54, Cambridge Univ. Press (1995) · Zbl 0878.58020 · doi:10.1017/CBO9780511809187
[29] G A Margulis, Positive harmonic functions on nilpotent groups, Dokl. Akad. Nauk SSSR 166 (1966) 1054 · Zbl 0187.38902
[30] H Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J. 66 (1992) 387 · Zbl 0780.30032 · doi:10.1215/S0012-7094-92-06613-0
[31] H Masur, S Tabachnikov, Rational billiards and flat structures (editors B Hasselblatt, A Katok), North-Holland (2002) 1015 · Zbl 1057.37034 · doi:10.1016/S1874-575X(02)80015-7
[32] K Matsuzaki, M Taniguchi, Hyperbolic manifolds and Kleinian groups, The Clarendon Press (1998) · Zbl 0892.30035
[33] C T McMullen, Prym varieties and Teichmüller curves, Duke Math. J. 133 (2006) 569 · Zbl 1099.14018 · doi:10.1215/S0012-7094-06-13335-5
[34] B Mohar, W Woess, A survey on spectra of infinite graphs, Bull. London Math. Soc. 21 (1989) 209 · Zbl 0645.05048 · doi:10.1112/blms/21.3.209
[35] M A Picardello, W Woess, Harmonic functions and ends of graphs, Proc. Edinburgh Math. Soc. 31 (1988) 457 · Zbl 0664.60075 · doi:10.1017/S0013091500037640
[36] P Przytycki, G Schmithüsen, F Valdez, Veech groups of Loch Ness monsters, Ann. Inst. Fourier (Grenoble) 61 (2011) 673 · Zbl 1266.32016 · doi:10.5802/aif.2625
[37] H Sato, Hausdorff dimension of the limit sets of classical Schottky groups, S\Burikaisekikenky\Busho K\Boky\Buroku (1996) 155 · Zbl 1044.30509
[38] K Schmidt, A cylinder flow arising from irregularity of distribution, Compositio Math. 36 (1978) 225 · Zbl 0388.28019
[39] M Schmoll, Veech groups for holonomy-free torus covers, J. Topol. Anal. 3 (2011) 521 · Zbl 1248.14034 · doi:10.1142/S1793525311000647
[40] J Smillie, C Ulcigrai, Beyond Sturmian sequences: Coding linear trajectories in the regular octagon, Proc. Lond. Math. Soc. 102 (2011) 291 · Zbl 1230.37021 · doi:10.1112/plms/pdq018
[41] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417 · Zbl 0674.57008 · doi:10.1090/S0273-0979-1988-15685-6
[42] R Treviño, On the ergodicity of flat surfaces of finite area, Geom. Funct. Anal. 24 (2014) 360 · Zbl 1300.37005 · doi:10.1007/s00039-014-0269-4
[43] S Troubetzkoy, Recurrence and periodic billiard orbits in polygons, Regul. Chaotic Dyn. 9 (2004) 1 · Zbl 1049.37024 · doi:10.1070/RD2004v009n01ABEH000259
[44] S Troubetzkoy, Typical recurrence for the Ehrenfest wind-tree model, J. Stat. Phys. 141 (2010) 60 · Zbl 1202.82008 · doi:10.1007/s10955-010-0026-5
[45] W A Veech, Boshernitzan’s criterion for unique ergodicity of an interval exchange transformation, Ergodic Theory Dynam. Systems 7 (1987) 149 · Zbl 0657.28012 · doi:10.1017/S0143385700003862
[46] W A Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989) 553 · Zbl 0676.32006 · doi:10.1007/BF01388890
[47] Y B Vorobets, G A Gal\(^{\prime}\)perin, A M Stëpin, Periodic billiard trajectories in polygons: generation mechanisms, Uspekhi Mat. Nauk 47 (1992) 9, 207 · Zbl 0777.58031 · doi:10.1070/RM1992v047n03ABEH000893
[48] W Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics 138, Cambridge Univ. Press (2000) · Zbl 0951.60002 · doi:10.1017/CBO9780511470967
[49] J C Yoccoz, Interval exchange maps and translation surfaces (editors M L Einsiedler, D A Ellwood, A Eskin, D Kleinbock, E Lindenstrauss, G Margulis, S Marmi, J C Yoccoz), Clay Math. Proc. 10, Amer. Math. Soc. (2010) 1 · Zbl 1195.37005
[50] A N Zemljakov, A B Katok, Topological transitivity of billiards in polygons, Mat. Zametki 18 (1975) 291 · Zbl 0315.58014
[51] A Zorich, Flat surfaces (editors P Cartier, B Julia, P Moussa, P Vanhove), Springer (2006) 437 · Zbl 1129.32012 · doi:10.1007/978-3-540-31347-2_13
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