# American Institute of Mathematical Sciences

2019, 14: 121-151. doi: 10.3934/jmd.2019005

## Dilation surfaces and their Veech groups

 1 Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG), Boite Courrier 7012, 8 Place Aurélie Nemours, 75013 Paris, France 2 Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany 3 Warwick Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

To the memory of William Veech

Received  August 29, 2018 Revised  February 12, 2019 Published  March 2019

We introduce a class of objects which we call 'dilation surfaces'. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples, and we define concepts related to these in order to motivate several questions and open problems. In particular we generalize the notion of Veech group to dilation surfaces, and we prove a structure result about these Veech groups.

Citation: Eduard Duryev, Charles Fougeron, Selim Ghazouani. Dilation surfaces and their Veech groups. Journal of Modern Dynamics, 2019, 14: 121-151. doi: 10.3934/jmd.2019005
##### References:
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##### References:
 [1] A. Boulanger, C. Fougeron and S. Ghazouani, Cascades in the dynamics of affine interval exchanges, to appear in Ergodic Theory, 2018.Google Scholar [2] X. Bressaud, P. Hubert and A. Maass, Persistence of wandering intervals in self-similar affine interval exchange transformations, Ergodic Theory Dynam. Systems, 30 (2010), 665-686. doi: 10.1017/S0143385709000418. Google Scholar [3] J. Bowman and S. Sanderson, Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces, arXiv: 1806.04129, (June, 2018).Google Scholar [4] R. Camelier and C. Gutierrez, Affine interval exchange transformations with wandering intervals, Ergodic Theory Dynam. Systems, 17 (1997), 1315-1338. doi: 10.1017/S0143385797097666. Google Scholar [5] E. Duryev and L. Monin, Twisted differentials, dilation surfaces and complex affine surfaces, in preparation, 2018.Google Scholar [6] W. M. Goldman, Geometric structures on manifolds and varieties of representations, in Geometry of Group Representations (Boulder, CO, 1987), Contemp. Math., 74, Amer. Math. Soc., Providence, RI, 1988, 169–198. doi: 10.1090/conm/074/957518. Google Scholar [7] R. C. Gunning, Affine and projective structures on Riemann surfaces, in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981, 225–244. Google Scholar [8] P. Hubert and T. A. Schmidt, Chapter 6 - An Introduction to Veech Surfaces, in Handbook of Dynamical Systems (ed. B. Hasselblatt and A. Katok), Vol. 1B, Elsevier B. V., Amsterdam, 2006, 501–526. doi: 10.1016/S1874-575X(06)80031-7. Google Scholar [9] G. Levitt, Feuilletages des surfaces, Ann. Inst. Fourier (Grenoble), 32 (1982), 179-217. doi: 10.5802/aif.875. Google Scholar [10] I. Liousse, Dynamique générique des feuilletages transversalement affines des surfaces, Bull. Soc. Math. France, 123 (1995), 493-516. doi: 10.24033/bsmf.2268. Google Scholar [11] R. Mandelbaum, Branched structures on Riemann surfaces, Trans. Amer. Math. Soc., 163 (1972), 261-275. doi: 10.1090/S0002-9947-1972-0288253-1. Google Scholar [12] R. Mandelbaum, Branched structures and affine and projective bundles on Riemann surfaces, Trans. Amer. Math. Soc., 183 (1973), 37-58. doi: 10.1090/S0002-9947-1973-0325958-9. Google Scholar [13] S. Marmi, P. Moussa and J.-C. Yoccoz, Affine interval exchange maps with a wandering interval, Proc. Lond. Math. Soc. (3), 100 (2010), 639-669. doi: 10.1112/plms/pdp037. Google Scholar [14] F. E. Prym, Zur Integration der gleichzeitigen Differentialgleichungen, J. Reine Angew. Math., 70 (1869), 354-362. doi: 10.1515/crll.1869.70.354. Google Scholar [15] W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, Edited by S. Levy, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997. Google Scholar [16] W. A. Veech, Flat surfaces, Amer. J. Math., 115 (1993), 589-689. doi: 10.2307/2375075. Google Scholar [17] W. A. Veech, Delaunay partitions, Topology, 36 (1997), 1-28. doi: 10.1016/0040-9383(96)00002-X. Google Scholar [18] W. A. Veech, Informal notes on flat surfaces, Unpublished course notes, 2008.Google Scholar [19] Ya. B. Vorobets, Plane structures and billiards in rational polygons: The Veech alternative, Uspekhi Mat. Nauk, 51 (1996), 3-42. doi: 10.1070/RM1996v051n05ABEH002993. Google Scholar [20] A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006, 437–583. doi: 10.1007/978-3-540-31347-2_13. Google Scholar
A translation surface of genus $2$
A 'dilation surface' of genus $2$ and a leaf of its horizontal foliation
A 'hyperbolic' closed leaf
The Franco-Russian slit construction
A Hopf torus and the basis of its homology
The double-chamber surface
Dilation cylinders of the double-chamber surface
The disco surface $\operatorname{D}_{a, b}$
An alternative representation of the disco surface
Cut-and-paste operation applied to the image of the double-chamber surface under the matrix $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$
A ribbon graph with two vertices
A cylinder decomposition of the surface of genus $2$
A dilation torus, which is not a Hopf torus
A dilation surface with a non-discrete set of holonomy vectors of saddle connections starting at the black point
An angular section in which all leaves are hyperbolic
Topological setting of the triangulation
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