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:
[1]

A. Boulanger, C. Fougeron and S. Ghazouani, Cascades in the dynamics of affine interval exchanges, to appear in Ergodic Theory, 2018.

[2]

X. BressaudP. 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.

[3]

J. Bowman and S. Sanderson, Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces, arXiv: 1806.04129, (June, 2018).

[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.

[5]

E. Duryev and L. Monin, Twisted differentials, dilation surfaces and complex affine surfaces, in preparation, 2018.

[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.

[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.

[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.

[9]

G. Levitt, Feuilletages des surfaces, Ann. Inst. Fourier (Grenoble), 32 (1982), 179-217. doi: 10.5802/aif.875.

[10]

I. Liousse, Dynamique générique des feuilletages transversalement affines des surfaces, Bull. Soc. Math. France, 123 (1995), 493-516. doi: 10.24033/bsmf.2268.

[11]

R. Mandelbaum, Branched structures on Riemann surfaces, Trans. Amer. Math. Soc., 163 (1972), 261-275. doi: 10.1090/S0002-9947-1972-0288253-1.

[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.

[13]

S. MarmiP. 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.

[14]

F. E. Prym, Zur Integration der gleichzeitigen Differentialgleichungen, J. Reine Angew. Math., 70 (1869), 354-362. doi: 10.1515/crll.1869.70.354.

[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.
[16]

W. A. Veech, Flat surfaces, Amer. J. Math., 115 (1993), 589-689. doi: 10.2307/2375075.

[17]

W. A. Veech, Delaunay partitions, Topology, 36 (1997), 1-28. doi: 10.1016/0040-9383(96)00002-X.

[18]

W. A. Veech, Informal notes on flat surfaces, Unpublished course notes, 2008.

[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.

[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.

show all references

References:
[1]

A. Boulanger, C. Fougeron and S. Ghazouani, Cascades in the dynamics of affine interval exchanges, to appear in Ergodic Theory, 2018.

[2]

X. BressaudP. 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.

[3]

J. Bowman and S. Sanderson, Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces, arXiv: 1806.04129, (June, 2018).

[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.

[5]

E. Duryev and L. Monin, Twisted differentials, dilation surfaces and complex affine surfaces, in preparation, 2018.

[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.

[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.

[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.

[9]

G. Levitt, Feuilletages des surfaces, Ann. Inst. Fourier (Grenoble), 32 (1982), 179-217. doi: 10.5802/aif.875.

[10]

I. Liousse, Dynamique générique des feuilletages transversalement affines des surfaces, Bull. Soc. Math. France, 123 (1995), 493-516. doi: 10.24033/bsmf.2268.

[11]

R. Mandelbaum, Branched structures on Riemann surfaces, Trans. Amer. Math. Soc., 163 (1972), 261-275. doi: 10.1090/S0002-9947-1972-0288253-1.

[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.

[13]

S. MarmiP. 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.

[14]

F. E. Prym, Zur Integration der gleichzeitigen Differentialgleichungen, J. Reine Angew. Math., 70 (1869), 354-362. doi: 10.1515/crll.1869.70.354.

[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.
[16]

W. A. Veech, Flat surfaces, Amer. J. Math., 115 (1993), 589-689. doi: 10.2307/2375075.

[17]

W. A. Veech, Delaunay partitions, Topology, 36 (1997), 1-28. doi: 10.1016/0040-9383(96)00002-X.

[18]

W. A. Veech, Informal notes on flat surfaces, Unpublished course notes, 2008.

[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.

[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.

Figure 1.  A translation surface of genus $ 2 $
Figure 2.  A 'dilation surface' of genus $ 2 $ and a leaf of its horizontal foliation
Figure 3.  A 'hyperbolic' closed leaf
Figure 5.  The Franco-Russian slit construction
Figure 4.  A Hopf torus and the basis of its homology
Figure 6.  The double-chamber surface
Figure 7.  Dilation cylinders of the double-chamber surface
Figure 8.  The disco surface $ \operatorname{D}_{a, b} $
Figure 9.  An alternative representation of the disco surface
Figure 10.  Cut-and-paste operation applied to the image of the double-chamber surface under the matrix $ \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} $
Figure 11.  A ribbon graph with two vertices
Figure 12.  A cylinder decomposition of the surface of genus $ 2 $
Figure 13.  A dilation torus, which is not a Hopf torus
Figure 14.  A dilation surface with a non-discrete set of holonomy vectors of saddle connections starting at the black point
Figure 15.  An angular section in which all leaves are hyperbolic
Figure 16.  Topological setting of the triangulation
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