Measured foliations on nonorientable surfaces

Danthony, Claude; Nogueira, Arnaldo

doi:10.24033/asens.1608

Cited by 35 publications

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“…Note that the pair .T 1 ; T 2 / can also be seen as a particular case of a linear involution, which was introduced by Danthony and Nogueira [3] in order to encode the first return map of a measured foliation on a transverse segment. See also [2].…”

Section: Constructions Of a Flat Surfacementioning

confidence: 99%

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Degenerations of quadratic differentials on ℂℙ¹

Boissy

2008

Geom. Topol.

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We describe the connected components of the complement of a natural "diagonal" of real codimension 1 in a stratum of quadratic differentials on CP 1 . We establish a natural bijection between the set of these connected components and the set of generic configurations that appear on such "flat spheres". We also prove that the stratum has only one topological end. Finally, we elaborate a necessary toolkit destined to evaluation of the Siegel-Veech constants.

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Section: Constructions Of a Flat Surfacementioning

confidence: 99%

“…Now the condition L 0 L implies that we can find a path 2 in the same homotopy class, such that the "-neighborhood of 2 is homeomorphic to a disk. Now joining carefully the endpoints of 2 to each sides of , we get a path 3 . By construction, we can use this path to contract the saddle connection .…”

Section: Reaching a Neighborhood Of The Principal Boundarymentioning

confidence: 99%

Degenerations of quadratic differentials on ℂℙ¹

Boissy

2008

Geom. Topol.

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“…Natural generalizations of interval exchange transformations were introduced by Danthony and Nogueira in [DN88,DN90] (see also [Nog89]) as cross sections of measured foliations on surfaces. They introduced the notion of linear involutions, as well as the notion of Rauzy induction on these maps.…”

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confidence: 99%

“…In this paper we use the definition of linear involution † (see Definition 2.1) proposed by Danthony and Nogueira (see [DN88,DN90]). † Let f be the involution of X × {0, 1} given by f (x, ε) = (x, 1 − ε) for (x, ε) ∈ X × {0, 1}.…”

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Dynamics and geometry of the Rauzy–Veech induction for quadratic differentials

Boissy

Lanneau

2009

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385708080565How to cite this article: CORENTIN BOISSY and ERWAN LANNEAU (2009). Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials.Abstract. Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy-Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmüller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely linear involutions. These maps are related to general measured foliations on surfaces (whether orientable or not). In this paper we are interested by such maps related to geodesic flow on (orientable) flat surfaces with Z/2Z linear holonomy. We relate geometry and dynamics of such maps to the combinatorics of generalized permutations. We study an analogue of the Rauzy-Veech induction and give an efficient combinatorial characterization of its attractors. We establish a natural bijection between the extended Rauzy classes of generalized permutations and connected components of the strata of meromorphic quadratic differentials with at most simple poles, which allows us, in particular, to classify the connected components of all exceptional strata.

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