# 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:

show all references

##### References:
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
 [1] Karol Mikula, Mariana Remešíková, Peter Novysedlák. Truss structure design using a length-oriented surface remeshing technique. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 933-951. doi: 10.3934/dcdss.2015.8.933 [2] John Franks, Michael Handel. Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface. Journal of Modern Dynamics, 2013, 7 (3) : 369-394. doi: 10.3934/jmd.2013.7.369 [3] Enrique R. Pujals, Federico Rodriguez Hertz. Critical points for surface diffeomorphisms. Journal of Modern Dynamics, 2007, 1 (4) : 615-648. doi: 10.3934/jmd.2007.1.615 [4] Lok Ming Lui, Chengfeng Wen, Xianfeng Gu. A conformal approach for surface inpainting. Inverse Problems & Imaging, 2013, 7 (3) : 863-884. doi: 10.3934/ipi.2013.7.863 [5] Michel Benaim, Morris W. Hirsch. Chain recurrence in surface flows. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 1-16. doi: 10.3934/dcds.1995.1.1 [6] Erica Clay, Boris Hasselblatt, Enrique Pujals. Desingularization of surface maps. Electronic Research Announcements, 2017, 24: 1-9. doi: 10.3934/era.2017.24.001 [7] Robert Brooks and Eran Makover. The first eigenvalue of a Riemann surface. Electronic Research Announcements, 1999, 5: 76-81. [8] Omri M. Sarig. Bernoulli equilibrium states for surface diffeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 593-608. doi: 10.3934/jmd.2011.5.593 [9] Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems & Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036 [10] Bum Ja Jin, Mariarosaria Padula. In a horizontal layer with free upper surface. Communications on Pure & Applied Analysis, 2002, 1 (3) : 379-415. doi: 10.3934/cpaa.2002.1.379 [11] Vera Mikyoung Hur. On the formation of singularities for surface water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1465-1474. doi: 10.3934/cpaa.2012.11.1465 [12] Kazuo Aoki, Pierre Charrier, Pierre Degond. A hierarchy of models related to nanoflows and surface diffusion. Kinetic & Related Models, 2011, 4 (1) : 53-85. doi: 10.3934/krm.2011.4.53 [13] Misha Bialy. On Totally integrable magnetic billiards on constant curvature surface. Electronic Research Announcements, 2012, 19: 112-119. doi: 10.3934/era.2012.19.112 [14] Alfonso Artigue. Anomalous cw-expansive surface homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3511-3518. doi: 10.3934/dcds.2016.36.3511 [15] Joachim Escher, Piotr B. Mucha. The surface diffusion flow on rough phase spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 431-453. doi: 10.3934/dcds.2010.26.431 [16] Giovanni Bellettini, Matteo Novaga, Giandomenico Orlandi. Eventual regularity for the parabolic minimal surface equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5711-5723. doi: 10.3934/dcds.2015.35.5711 [17] E.B. Pitman, C.C. Nichita, A.K. Patra, A.C. Bauer, M. Bursik, A. Webb. A model of granular flows over an erodible surface. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 589-599. doi: 10.3934/dcdsb.2003.3.589 [18] Octavian G. Mustafa. On isolated vorticity regions beneath the water surface. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1523-1535. doi: 10.3934/cpaa.2012.11.1523 [19] Georgi I. Kamberov. Recovering the shape of a surface from the mean curvature. Conference Publications, 1998, 1998 (Special) : 353-359. doi: 10.3934/proc.1998.1998.353 [20] Alfonso Artigue. Robustly N-expansive surface diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2367-2376. doi: 10.3934/dcds.2016.36.2367

2018 Impact Factor: 0.295