# 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:
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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