Translation surfaces reading group

Schedule

• Tuesday 2024-10-08 at 10:30 (Masoud, based on [AthreyaMasur 2024])
  • three models
    • gluing sides of polygons
    • manifold with flat changes of coordinates
    • holomorphic 1-form on a Riemann surface
  • some examples, singularities, cone angles, curvature, genus
• Wednesday 2024-10-16 at 14:30 (Thierry)
  • catch-up session for those who missed the first talk, motivated by billiards
• Tuesday 2024-10-29 at 13:30 (Masoud, based on [AthreyaMasur 2024])
  • equivalence between the 3 models
• Tuesday 2024-11-12 at 10:30 (Masoud, based on [AthreyaMasur 2024])
  • Moduli space of translation surfaces
  • we meet at 10:00 to debrief previous sessions
• Tuesday 2024-11-19 at 10:30 (Masoud, based on [AthreyaMasur 2024])
  • Measure on the moduli space of translation surfaces
  • we meet at 10:00 to debrief previous sessions
• Wednesday 2024-12-02 at 13:30 (Masoud, based on [AthreyaMasur 2024])
  • finiteness of the MSV measure
• Tuesday 2024-12-17 at 10:30 (Thierry)
  • several frameworks of dynamical systems (measurable, measured, topological, smooth, algebraic, ...), and several times (discrete, continuous, group actions, ...)
  • some examples (rotation, doubling map, vector fields, translation flow, interval exchange transformations, Arnold's cat map, SL(2,R) action on the strata)
  • measured dynamics
    • proof of the Poincaré reccurrence theorem and first return map
• Tuesday 2024-12-17 at 15:00 (Thierry)
  • measured dynamics (continued)
    • Birkhoff theorem and ergodicity
    • strong and weak mixing
  • topological dynamics
    • minimality
    • the Choquet simplex of invariant measures as a way to measure the diversity of behaviours in a topological dynamical system (following chapter 7.2 of [FerencziMonteil 2010], especially the proof of Theorem 7.2.4)
• Beginning of 2025 (likely 7th of January, to be confirmed)
  • debrief of chapters 2 and 3 of [AthreyaMasur 2024]: everybody is asked to reread the first 3 chapters and note what is unclear to them, so that we can fix that and continue on a firm basis.
  • dynamics in the context of translation surfaces

References

Translation surfaces, low-tech

• [AthreyaMasur 2024] Athreya and Masur, Translation surfaces, 2024 https://library.gift/main/974AE82D375AD57D90FFA34AF735795E
• [Zorich 2006] Zorich, Flat surfaces, 2006 https://webusers.imj-prg.fr/~anton.zorich/Papers/zorich_leshouches.pdf
• [Massart 2021] Massart, A short introduction to translation surfaces, Veech surfaces, and Teichmueller dynamics, 2021 https://arxiv.org/abs/2107.11581
• [BudChen 2020] Bud and Chen, Moduli of Differentials and Teichmüller Dynamics, 2020 https://arxiv.org/abs/2004.05762
• [Masur 2006] Masur, Ergodic theory of translation surfaces, 2006 https://library.gift/main/23A4BAC8CA3B74D1A7469182D86FFD2C
• [Eskin 2006] Eskin, Counting problems in moduli space, 2006 same link as [Masur 2006]

Translation surfaces, high-tech

• Lyapunov exponents
  • [Viana 2007] Viana, Lyapunov exponents of Teichmüller flows, 2007
  • [ForniMatheus 2014] Forni and Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, 2014 https://www.aimsciences.org/article/doi/10.3934/jmd.2014.8.271
  • [Forni 2023] Forni, Effective Unique Ergodicity and Weak Mixing of Translation Flows, 2023 https://arxiv.org/abs/2311.02714
• Classification of orbit closures
  • [Zorich 2015] Zorich, The Magic Wand Theorem of A. Eskin and M. Mirzakhani, 2015 https://arxiv.org/abs/1502.05654
  • [Wright 2016] Wright, From rational billiards to dynamics on moduli spaces, 2016 https://arxiv.org/abs/1504.08290

Mapping class group

• [FarbMargalit 2012] Farb and Margalit, A Primer on Mapping Class Groups, 2012

Interval exchange transformations

• [Yoccoz 2010] Yoccoz, Interval exchange maps and translation surfaces, 2010 https://www.college-de-france.fr/sites/default/files/documents/jean-christophe-yoccoz/UPL15305_PisaLecturesJCY2007.pdf

Veech surfaces

• [HubertSchmidt 2006] Hubert and Schmidt, An introduction to Veech surfaces, 2006 same link as [Masur 2006]
• [Schmithüsen 2006] Schmithüsen, Examples for Veech groups of origamis, 2006
• [McMullen 2023] McMullen, Billiards and Teichmüller curves, 2023 https://people.math.harvard.edu/~ctm/papers/home/text/papers/tsurv/tsurv.pdf

Square-tiled surfaces

• [DelecroixGendronMatheus 2021] Delecroix and Gendron and Matheus, Combinatorics of Square-Tiled Surfaces and Geometry of Moduli Spaces, 2021 https://indico.ictp.it/event/9617/session/2/contribution/3/material/4/0.pdf
• [Matheus 2022] Matheus, Three lectures on square-tiled surfaces, 2022 https://if-summer2018.sciencesconf.org/data/pages/origamis_Grenoble_matheus_3.pdf

Symbolic dynamics

• [FerencziMonteil 2010] Ferenczi and Monteil, Infinite words with uniform frequencies, and invariant measures, 2010 https://library.gift/main/1A2CED9BED160AC939351015DBA389DA

Topics to be dealt with (tentative, feel free to propose more or give more details)

• general background
  • rigidity in complex analysis, pants, uniformization, Riemann-Roch, ... (Meltem ?)
  • Teichmüller Theory, quasiconformal maps, quadratic forms, ...
  • mapping class group, generators, growth, including that of non-orientable surfaces, [FarbMargalit 2012] (Pallavi)
  • topological dynamics and ergodic theory (Thierry)
• related to translation surfaces
  • connnected components of strata [KontsevichZorich 2002]
  • Masur criterion for unique ergodicity, Kerkhoff-Masur-Smilie
  • Veech group and Veech/lattices surfaces
    • Haar measure on SL(2,R)
    • properties of Veech groups
    • Veech alternative
    • classification in genus 2
  • square tiled surfaces
    • classification in H(2)
  • connection with tilings
  • interval exchange transformations and Rauzy-Veech induction (Carole)
  • volume estimation à la Eskin-Okounkov (Mingkun)
  • orbit closures and magic-wand theorem

Location

We meet at LIPN, room B107

Contact

Send an email to thierry.monteil@lipn.univ-paris13.fr if you want to join, to receive the annoucements, if you have wishes for the next topics, references to share, etc. Attach your SSH public key if you want to modify this page yourself.