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The invariant measures of some infinite interval exchange maps

W Patrick Hooper

Geometry & Topology 19 (2015) 1895–2038
Bibliography
1 J Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs 50, Amer. Math. Soc. (1997) MR1450400
2 J Aaronson, H Nakada, O Sarig, R Solomyak, Invariant measures and asymptotics for some skew products, Israel J. Math. 128 (2002) 93 MR1910377
3 C D Aliprantis, R Tourky, Cones and duality, Graduate Studies in Mathematics 84, Amer. Math. Soc. (2007) MR2317344
4 A F Beardon, B Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974) 1 MR0333164
5 F Bonahon, Geodesic laminations on surfaces, from: "Laminations and foliations in dynamics, geometry and topology" (editors M Lyubich, J W Milnor, Y N Minsky), Contemp. Math. 269, Amer. Math. Soc. (2001) 1 MR1810534
6 J P Bowman, The complete family of Arnoux–Yoccoz surfaces, Geom. Dedicata 164 (2013) 113 MR3054619
7 J P Bowman, F Valdez, Wild singularities of flat surfaces, Israel J. Math. 197 (2013) 69 MR3096607
8 D Burago, Y Burago, S Ivanov, A course in metric geometry, Graduate Studies in Mathematics 33, Amer. Math. Soc. (2001) MR1835418
9 R Chamanara, Affine automorphism groups of surfaces of infinite type, from: "In the tradition of Ahlfors and Bers, III" (editors W Abikoff, A Haas), Contemp. Math. 355, Amer. Math. Soc. (2004) 123 MR2145060
10 R Chamanara, F P Gardiner, N Lakic, A hyperelliptic realization of the horseshoe and baker maps, Ergodic Theory Dynam. Systems 26 (2006) 1749 MR2279264
11 Y Cheung, A Eskin, Unique ergodicity of translation flows, from: "Partially hyperbolic dynamics, laminations, and Teichmüller flow" (editors G Forni, M Lyubich, C Pugh, M Shub), Fields Inst. Commun. 51, Amer. Math. Soc. (2007) 213 MR2388697
12 J P Conze, Recurrence, ergodicity and invariant measures for cocycles over a rotation, from: "Ergodic theory" (editor I Assani), Contemp. Math. 485, Amer. Math. Soc. (2009) 45 MR2553209
13 J P Conze, K Frączek, Cocycles over interval exchange transformations and multivalued Hamiltonian flows, Adv. Math. 226 (2011) 4373 MR2770454
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 MR2901357
15 J P Conze, M Keane, Ergodicité d'un flot cylindrique, from: "Séminaire de Probabilités, I", Dépt. Math. Informat., Univ. Rennes (1976) MR0584020
16 B D Craven, J J Koliha, Generalizations of Farkas' theorem, SIAM J. Math. Anal. 8 (1977) 983 MR0471302
17 V Delecroix, P Hubert, S Lelièvre, Diffusion for the periodic wind-tree model, Ann. Sci. Éc. Norm. Supér. 47 (2014) 1085 MR3297155
18 J Farkas, Über die Theorie der Einfachen Ungleichungen, J. Reine Angew. Math. 124 (1902) 1
19 K Fraczek, C Ulcigrai, Non-ergodic $\mathbb{Z}$–periodic billiards and infinite translation surfaces, Invent. Math. 197 (2014) 241 MR3232007
20 J Hardy, J Weber, Diffusion in a periodic wind-tree model, J. Math. Phys. 21 (1980) 1802 MR575616
21 W P Hooper, Dynamics on an infinite surface with the lattice property, arXiv:0802.0189
22 W P Hooper, Grid graphs and lattice surfaces, Int. Math. Res. Not. 2013 (2013) 2657 MR3071661
23 W P Hooper, An infinite surface with the lattice property I: Veech groups and coding geodesics, Trans. Amer. Math. Soc. 366 (2014) 2625 MR3165649
24 W P Hooper, P Hubert, B Weiss, Dynamics on the infinite staircase, Discrete Contin. Dyn. Syst. 33 (2013) 4341 MR3038066
25 W P Hooper, B Weiss, Generalized staircases: Recurrence and symmetry, Ann. Inst. Fourier (Grenoble) 62 (2012) 1581 MR3025751
26 P Hubert, S Lelièvre, S Troubetzkoy, The Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion, J. Reine Angew. Math. 656 (2011) 223 MR2818861
27 P Hubert, G Schmithüsen, Infinite translation surfaces with infinitely generated Veech groups, J. Mod. Dyn. 4 (2010) 715 MR2753950
28 A Katok, B Hasselblatt, Introduction to the modern theory of dynamical systems, Encycl. Math. and Appl. 54, Cambridge Univ. Press (1995) MR1326374
29 G A Margulis, Positive harmonic functions on nilpotent groups, Dokl. Akad. Nauk SSSR 166 (1966) 1054 MR0222217
30 H Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J. 66 (1992) 387 MR1167101
31 H Masur, S Tabachnikov, Rational billiards and flat structures, from: "Handbook of dynamical systems, Vol. 1A" (editors B Hasselblatt, A Katok), North-Holland (2002) 1015 MR1928530
32 K Matsuzaki, M Taniguchi, Hyperbolic manifolds and Kleinian groups, The Clarendon Press (1998) MR1638795
33 C T McMullen, Prym varieties and Teichmüller curves, Duke Math. J. 133 (2006) 569 MR2228463
34 B Mohar, W Woess, A survey on spectra of infinite graphs, Bull. London Math. Soc. 21 (1989) 209 MR986363
35 M A Picardello, W Woess, Harmonic functions and ends of graphs, Proc. Edinburgh Math. Soc. 31 (1988) 457 MR969074
36 P Przytycki, G Schmithüsen, F Valdez, Veech groups of Loch Ness monsters, Ann. Inst. Fourier (Grenoble) 61 (2011) 673 MR2895069
37 H Sato, Hausdorff dimension of the limit sets of classical Schottky groups, Sūrikaisekikenkyūsho Kōkyūroku (1996) 155 MR1483414
38 K Schmidt, A cylinder flow arising from irregularity of distribution, Compositio Math. 36 (1978) 225 MR515047
39 M Schmoll, Veech groups for holonomy-free torus covers, J. Topol. Anal. 3 (2011) 521 MR2887675
40 J Smillie, C Ulcigrai, Beyond Sturmian sequences: Coding linear trajectories in the regular octagon, Proc. Lond. Math. Soc. 102 (2011) 291 MR2769116
41 W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417 MR956596
42 R Treviño, On the ergodicity of flat surfaces of finite area, Geom. Funct. Anal. 24 (2014) 360 MR3177386
43 S Troubetzkoy, Recurrence and periodic billiard orbits in polygons, Regul. Chaotic Dyn. 9 (2004) 1 MR2058893
44 S Troubetzkoy, Typical recurrence for the Ehrenfest wind-tree model, J. Stat. Phys. 141 (2010) 60 MR2720043
45 W A Veech, Boshernitzan's criterion for unique ergodicity of an interval exchange transformation, Ergodic Theory Dynam. Systems 7 (1987) 149 MR886376
46 W A Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989) 553 MR1005006
47 Y B Vorobets, G A Gal′perin, A M Stëpin, Periodic billiard trajectories in polygons: generation mechanisms, Uspekhi Mat. Nauk 47 (1992) 9, 207 MR1185299
48 W Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics 138, Cambridge Univ. Press (2000) MR1743100
49 J C Yoccoz, Interval exchange maps and translation surfaces, from: "Homogeneous flows, moduli spaces and arithmetic" (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 MR2648692
50 A N Zemljakov, A B Katok, Topological transitivity of billiards in polygons, Mat. Zametki 18 (1975) 291 MR0399423
51 A Zorich, Flat surfaces, from: "Frontiers in number theory, physics, and geometry, I" (editors P Cartier, B Julia, P Moussa, P Vanhove), Springer (2006) 437 MR2261104