Abstract
This section serves as a general-purpose introduction to all other sections of this memoir. In particular, we’ll always assume familiarity with the content of this section in subsequent discussions.
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- 1.
By precomposition with coordinate charts.
- 2.
Since translations are particular cases of biholomorphisms.
- 3.
We define \(\omega \) by locally pulling-back dz via the charts: this gives a globally defined Abelian differential because the changes of coordinates are translations and, hence, dz is invariant under changes of coordinates.
- 4.
Alternatively, this fact can be derived from Poincaré-Hopf index theorem applied to the vector field given by the vertical direction at all points of \(L-\{p\}\).
- 5.
A translation flow is obtained by moving (almost all) points of a translation surface in a fixed direction.
- 6.
I.e., the angle of reflection equals the angle of incidence.
- 7.
It is possible to prove that \(\mathcal {L}(\kappa )\) is non-empty whenever \(\sum \limits _{l=1}^{\sigma } k_l=2g-2\).
- 8.
Here, we use the developing map to put a natural topology on \(\mathcal {TH}(\kappa )\). More concretely, given \(\omega \in \mathcal {L}(\kappa )\), \(p_0\in \text {div}(\omega )\), an universal cover \(p:\widetilde{M}\rightarrow M\) and \(P_1\in p^{-1}(p_0)\), we have a developing map \(D_{\omega }:(\widetilde{M}, P_0)\rightarrow (\mathbb {C}, 0)\) determining completely the translation structure \((M,\omega )\). The injective map \(\omega \mapsto D_{\omega }\) gives a copy of \(\mathcal {L}(\omega )\) inside the space \(C^0(\widetilde{M},\mathbb {C})\) of complex-valued continuous functions of \(\widetilde{M}\). In particular, the compact-open topology of \(C^0(\widetilde{M}, \mathbb {C})\) induces natural topologies on \(\mathcal {L}(\kappa )\) and \(\mathcal {TH}(\kappa )\).
- 9.
Via the so-called Gauss-Manin connection.
- 10.
In general, \(\mathcal {H}(\kappa )\) are not manifolds: for example, the moduli space \(\mathcal {H}(0)\) of flat torii is \(GL^+(2,\mathbb {R})/SL(2,\mathbb {Z})\).
- 11.
A slight modification of the notion of Rauzy classes introduced by Rauzy [60] in his study of i.e.t.’s.
- 12.
Rauzy classes are complicated objects: the cardinalities of the largest Rauzy classes associated to \(\mathcal {H}_2\), \(\mathcal {H}_3\), \(\mathcal {H}_4\) and \(\mathcal {H}_5\) are 15, 2177, 617401 and 300296573.
- 13.
It is possible to prove that the value \(\Phi (M,\omega )\in \mathbb {Z}/2\mathbb {Z}\) independs of the choice.
- 14.
The sets \(\mathcal {H}^{(a)}(\kappa )\) are “hyperboloids” inside \(\mathcal {H}(\kappa )\): indeed, this follows from the fact that \(A_{\kappa }(\omega ) = \frac{i}{2}(\sum \limits _{n=1}^g (A_n \overline{B_n} - \overline{A_n} B_n)\) where \(A_n=\int _{\alpha _n}\omega \) and \(B_n=\int _{\beta _n}\omega \) are the periods of \(\omega \) with respect to a canonical symplectic basis \(\{\alpha _n, \beta _n\}_{n=1}^g\) of \(H_1(M,\mathbb {R})\).
- 15.
Coming from Poincaré recurrence theorem.
- 16.
A similar definition can be performed over the action of \(SL(2,\mathbb {R})\) and, by a slight abuse of notation, we shall also call “Kontsevich-Zorich cocycle” the resulting object.
- 17.
For example, we can take \(\Vert .\Vert \) to be the so-called Hodge norm, see [28].
- 18.
This reflects the fact that the eigenvalues of a symplectic matrix comes in pairs of the form \(\theta \) and \(1/\theta \).
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Matheus Silva Santos, C. (2018). Introduction. In: Dynamical Aspects of Teichmüller Theory. Atlantis Studies in Dynamical Systems, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-92159-4_1
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DOI: https://doi.org/10.1007/978-3-319-92159-4_1
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