I am trying to learn more about dilation surfaces, in the context of dynamical systems, but there is not much information about them online. One definition I see is:
A dilation surface is a collection of polygons with sides glued together in parallel opposite pairs by maps that are the composition of dilations by positive real factors and translations.
This is confusing to me, but I am pretty sure it can be explained a little simpler. Does anyone have any information on this, or could perhaps provide a detailed explanation of what these actually are?
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$\begingroup$This notion was indeed not studied substantially by the mathematical community. So it is normal to not find a lot of survey about it.
I think you should first look at translation surface which are studied extensively and have a wikipedia page. The idea is to unfold a surface into a polygon, and if you want to do it backward to glue the parallel boundaries of polygon to have a surface. The most famous example is the torus which is a square with opposites sides identified by translation.
Now you can broaden your definition and allow not only translation but also dilation. This give you the dilation surfaces. You can read the paper of E. Duryev, C. Fougeron, S. Ghazouani which gives a good presentation on this subject. You can also take a look of the slides of the presentation of Jane Wang "The Realization Problem for Twisted Quadratic Differentials (Dilation Surfaces)".
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