We present some works of Corinna Ulcigrai closely related to Diophantine approximations and generalizing classical notions to the context of interval exchange maps, translation surfaces and Teichmüller dynamics.
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Figure 4. Rauzy diagram for $ d = 6 $ (using the same set of notations [46]). This figure is an extract of the Master thesis of Corinna Ulcigrai, Pisa, 2002
[1] |
M. Artigiani, L. Marchese and C. Ulcigrai, The Lagrange spectrum of a Veech surface has a Hall ray, Groups Geom. Dyn., 10 (2016), 1287-1337.
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[2] |
M. Artigiani, L. Marchese and C. Ulcigrai, Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces, Ergodic Theory Dynam. Systems, 40 (2020), 2017-2072.
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M. Boshernitzan and V. Delecroix, From a packing problem to quantitative recurrence in $[0, 1]$ and the Lagrange spectrum of interval exchanges, Discrete Anal., (2017), 25 pp.
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A. D. Brjuno, Analytic form of differential equations. I (Russian), Trudy Moskov. Mat. Obšč., 25 (1971), 119-262.
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A. D. Brjuno, Analytic form of differential equations. II (Russian), Trudy Moskov. Mat. Obšč., 26 (1972), 199-239.
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A. Bufetov and G. Forni, Limit theorems for horocycle flows, Ann. Sci. Éc. Norm. Supér., 47 (2014), 891-903.
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A. Bufetov, Limit theorems for translation flows, Ann. of Math., 179 (2014), 431-499.
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S. Marmi, D. H. Kim and L. Marchese, Long hitting time for translation flows and L-shaped billiards, J. Mod. Dyn., 14 (2019), 291-353.
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A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL(2, $\mathbb R$) action on moduli space, Publ. Math. Inst. Hautes Études Sci., 127 (2018), 95–324.
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A. Fathi, F. Laudenbach and V. Poénaru, Thurston's Work on Surfaces, Mathematical Notes, 48, Princeton University Press, Princeton, NJ, 2012.
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G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math., 146 (1997), 295-344.
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G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math., 155 (2002), 1-103.
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S. Ghazouani, Local rigidity for periodic generalised interval exchange transformations, Invent. Math., 226 (2021), 467-520.
doi: 10.1007/s00222-021-01051-3.![]() ![]() ![]() |
[17] |
S. Ghazouani and C. Ulcigrai, A priori bounds for giets, affine shadows and rigidity of foliations in genus 2, preprint, arXiv: 2106.03529, 2021.
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[18] |
M. Hall Jr., On the sum and product of continued fractions, Ann. of Math., 48 (1947), 966-993.
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5$^{th}$ edition, The Clarendon Press, Oxford University Press, New York, 1979.
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M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math, 49 (1979), 5-233.
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P. Hubert, S. Lelièvre, L. Marchese and C. Ulcigrai, The Lagrange spectrum of some square-tiled surfaces, Israel J. Math., 225 (2018), 553-607.
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[22] |
P. Hubert, L. Marchese and C. Ulcigrai, Lagrange spectra in Teichmüller dynamics via renormalization, Geom. Funct. Anal., 25 (2015), 180-255.
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M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31.
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M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196.
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H. B. Keynes and D. Newton, A "minimal", non-uniquely ergodic interval exchange transformation, Math. Z., 148 (1976), 101-105.
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A. Y. Khinchin, Continued Fractions, University of Chicago Press, Chicago, Ill.-London, 1964.
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D. H. Kim and S. Marmi, The recurrence time for interval exchange maps, Nonlinearity, 21 (2008), 2201-2210.
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D. Lima, C. Matheus, C. G. Moreira and S. Romaña, Classical and Dynamical Markov and Lagrange Spectra—Dynamical, Fractal and Arithmetic Aspects, World Scientific Publishing C. Pte. Ltd., Hackensack, NJ, 2021.
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S. Marmi, P. Moussa and J.-C. Yoccoz, Some properties of real and complex Brjuno functions, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006,601–623.
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[30] |
S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps, J. Amer. Math. Soc., 18 (2005), 823-872.
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S. Marmi, P. Moussa and J.-C. Yoccoz, Affine interval exchange maps with a wandering interval, Proc. Lond. Math. Soc., 100 (2010), 639-669.
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S. Marmi, P. Moussa and J.-C. Yoccoz, Linearization of generalized interval exchange maps, Ann. of Math., 176 (2012), 1583-1646.
doi: 10.4007/annals.2012.176.3.5.![]() ![]() ![]() |
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S. Marmi, C. Ulcigrai and J.-C. Yoccoz, On Roth type conditions, duality and central Birkhoff sums for i.e.m., Astérisque, 416 (2020), 65-132.
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[34] |
S. Marmi and J.-C. Yoccoz, Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps, Comm. Math. Phys., 344 (2016), 117-139.
doi: 10.1007/s00220-016-2624-9.![]() ![]() ![]() |
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H. Masur, Interval exchange transformations and measured foliations, Ann. of Math., 115 (1982), 169-200.
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K. F. Roth, Rational approximations to algebraic numbers, Mathematika, 2 (1955), 1-20.
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C. Series, The modular surface and continued fractions, J. London Math. Soc., 31 (1985), 69-80.
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J. Smillie and B. Weiss, Minimal sets for flows on moduli space, Israel J. Math., 142 (2004), 249-260.
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W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., 115 (1982), 201-242.
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W. A. Veech, Boshernitzan's criterion for unique ergodicity of an interval exchange transformation, Ergodic Theory Dynam. Systems, 7 (1987), 149-153.
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Y. B. Vorobets, Planar structures and billiards in rational polygons: The Veech alternative, Russian Math. Surveys, 51 (1996), 779-817.
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J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition Diophantienne, in Ann. Sci. École Norm. Sup. (4), 17 (1984), 333–359.
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J.-C. Yoccoz, Linéarisation des germes de difféomorphismes holomorphes de (C, 0), C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 55-58.
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J.-C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, 231 (1995).
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[45] |
J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms, in Dynamical Systems and Small Divisors (Cetraro, 1998), Lecture Notes in Math., 1784, Fond. CIME/CIME Found. Subser., Springer, Berlin, 2002,125–173.
doi: 10.1007/978-3-540-47928-4_3.![]() ![]() ![]() |
[46] |
J.-C. Yoccoz, Continued fraction algorithms for interval exchange maps: An introduction, Frontiers in Number Theory, Physics and Geometry, 1 (2006), 401-435.
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J.-C. Yoccoz, Interval exchange maps and translation surfaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 1–69.
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A. Zorich, Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477-1499.
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A. Zorich, How do the leaves of a closed 1-form wind around a surface?, in Pseudoperiodic Topology, Amer. Math. Soc. Transl. Ser. 2,197, Adv. Math. Sci., 46, Amer. Math. Soc., Providence, RI, 1999.
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A. Zorich, Flat surfaces, Frontiers in Number Theory, Physics, and Geometry, 1 (2006), 437-583.
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The Brjuno function
Rauzy diagram for 2 and 3 intervals
Rauzy diagrams for 4 intervals and genus 2
Rauzy diagram for