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Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics
Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces
1. | Natural Science Division, Pepperdine University, 24255 Pacific Coast Highway, Malibu, CA 90263, USA |
2. | Department of Mathematics, Kidder Hall 368, Oregon State University, Corvallis, OR 97331, USA |
A homothety surface can be assembled from polygons by identifying their edges in pairs via homotheties, which are compositions of translation and scaling. We consider linear trajectories on a $ 1 $-parameter family of genus-$ 2 $ homothety surfaces. The closure of a trajectory on each of these surfaces always has Hausdorff dimension $ 1 $, and contains either a closed loop or a lamination with Cantor cross-section. Trajectories have cutting sequences that are either eventually periodic or eventually Sturmian. Although no two of these surfaces are affinely equivalent, their linear trajectories can be related directly to those on the square torus, and thence to each other, by means of explicit functions. We also briefly examine two related families of surfaces and show that the above behaviors can be mixed; for instance, the closure of a linear trajectory can contain both a closed loop and a lamination.
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Affine lamination spaces for surfaces, Pacific J. Math., 154 (1992), 87-101.
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M. Herman,
Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. maths. de l'I.H.É.S., 49 (1979), 5-233.
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[13] |
S. Janson and A. Öberg,
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J. Liouville, Sur quelques séries et produits infinis, J. math. pures et appl. 2e série, 2 (1857), 433-440. Google Scholar |
[18] |
M. Laurent and A. Nogueira,
Rotation number of contracted rotations, J. Mod. Dyn., 12 (2018), 175-191.
doi: 10.3934/jmd.2018007. |
[19] |
S. Marmi, P. Moussa and J.-C. Yoccoz,
Affine interval exchange maps with a wandering interval, Proc. London Math. Soc., 100 (2010), 639-669.
doi: 10.1112/plms/pdp037. |
[20] |
W. Thurston,
On the geometry and dynamics of diffeomorphisms of surfaces, Bull. A.M.S., 19 (1988), 417-431.
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show all references
References:
[1] |
D. Bailey and R. Crandall,
Random generators and normal numbers, Exp. Math., 11 (2002), 527-546.
doi: 10.1080/10586458.2002.10504704. |
[2] |
S. Bates and A. Norton,
On sets of critical values in the real line, Duke Math. J., 83 (1996), 399-413.
doi: 10.1215/S0012-7094-96-08313-1. |
[3] |
A. Besicovitch and S. Taylor,
On the complementary intervals of a linear closed set of zero Lebesgue measure, J. London Math. Soc., 29 (1954), 449-459.
doi: 10.1112/jlms/s1-29.4.449. |
[4] |
A. Boulanger, C. Fougeron and S. Ghazouani, Cascades in the dynamics of affine interval exchange transformations, Erg. Th. Dyn. Sys., (2019), 1–25.
doi: 10.1017/etds.2018.141. |
[5] |
Y. Bugeaud,
Dynamique de certaines applications contractantes, linéaires par morceaux, sur [0, 1[, C.R.A.S. Série I, 317 (1993), 575-578.
|
[6] |
Y. Bugeaud and J.-P. Conze,
Calcul de la dynamique de transformations linéaires contractantes mod 1 et arbre de Farey, Acta Arith., 88 (1999), 201-218.
doi: 10.4064/aa-88-3-201-218. |
[7] |
R. Coutinho, Dinâmica Simbólica Linear, Ph. D. Thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa, 1999. Google Scholar |
[8] |
R. Coutinho, B. Fernandez, R. Lima and A. Meyroneinc,
Discrete time piecewise affine models of genetic regulatory networks, J. Math. Biol., 52 (2006), 524-570.
doi: 10.1007/s00285-005-0359-x. |
[9] |
E. J. Ding and P. C. Hemmer,
Exact treatment of mode locking for a piecewise linear map, J. Stat. Phys., 46 (1987), 99-110.
doi: 10.1007/BF01010333. |
[10] |
E. Duryev, C. Fougeron and S. Ghazouani,
Dilation surfaces and their Veech groups, J. Mod. Dyn., 14 (2019), 121-151.
doi: 10.3934/jmd.2019005. |
[11] |
A. Hatcher and U. Oertel,
Affine lamination spaces for surfaces, Pacific J. Math., 154 (1992), 87-101.
doi: 10.2140/pjm.1992.154.87. |
[12] |
M. Herman,
Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. maths. de l'I.H.É.S., 49 (1979), 5-233.
|
[13] |
S. Janson and A. Öberg,
A piecewise contractive dynamical system and election methods, Bull. Soc. Math. de France, 147 (2019), 395-441.
doi: 10.24033/bsmf.2787. |
[14] |
A. Khinchin, Continued Fractions, Courier Corporation, 1964. |
[15] |
L. Kuipers and H. Neiderreiter, Uniform Distribution of Sequences, John Wiley & Sons, 1974. |
[16] |
I. Liousse,
Dynamique générique des feuilletages transversalement affines des surfaces, Bull. S.M.F., 123 (1995), 493-516.
doi: 10.24033/bsmf.2268. |
[17] |
J. Liouville, Sur quelques séries et produits infinis, J. math. pures et appl. 2e série, 2 (1857), 433-440. Google Scholar |
[18] |
M. Laurent and A. Nogueira,
Rotation number of contracted rotations, J. Mod. Dyn., 12 (2018), 175-191.
doi: 10.3934/jmd.2018007. |
[19] |
S. Marmi, P. Moussa and J.-C. Yoccoz,
Affine interval exchange maps with a wandering interval, Proc. London Math. Soc., 100 (2010), 639-669.
doi: 10.1112/plms/pdp037. |
[20] |
W. Thurston,
On the geometry and dynamics of diffeomorphisms of surfaces, Bull. A.M.S., 19 (1988), 417-431.
doi: 10.1090/S0273-0979-1988-15685-6. |
[21] |
W. Veech,
Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Inv. Math., 97 (1989), 553-583.
doi: 10.1007/BF01388890. |
[22] |
P. Veerman,
Symbolic dynamics of order-preserving orbits, Physica D: Nonlinear Phenomena, 29 (1987), 191-201.
doi: 10.1016/0167-2789(87)90055-8. |








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