Some arithmetical aspects of renormalization in Teichmüller dynamics: on the occasion of Corinna Ulcigrai winning the Brin Prize. (English) Zbl 1490.11068
Summary: 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.
MSC:
| 11J06 | Markov and Lagrange spectra and generalizations |
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
| 37E05 | Dynamical systems involving maps of the interval |
| 37E20 | Universality and renormalization of dynamical systems |
Keywords:
Diophantine conditions; Roth type; interval exchange maps; Birkhoff sums; deviation of ergodic averages; Lagrange spectrum; Veech surfaces; Hall rayBiographic References:
Ulcigrai, CorinnaReferences:
| [1] | M. Artigiani; L. Marchese; C. Ulcigrai, The Lagrange spectrum of a Veech surface has a Hall ray, Groups Geom. Dyn., 10, 1287-1337 (2016) · Zbl 1416.11122 · doi:10.4171/GGD/384 |
| [2] | M. Artigiani; L. Marchese; C. Ulcigrai, Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces, Ergodic Theory Dynam. Systems, 40, 2017-2072 (2020) · Zbl 1446.37010 · doi:10.1017/etds.2018.143 |
| [3] | A. Avila; M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198, 1-56 (2007) · Zbl 1143.37001 · doi:10.1007/s11511-007-0012-1 |
| [4] | M. Boshernitzan, A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J., 52, 723-752 (1985) · Zbl 0602.28009 · doi:10.1215/S0012-7094-85-05238-X |
| [5] | 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. · Zbl 1404.37042 |
| [6] | A. D. Brjuno, Analytic form of differential equations. I (Russian), Trudy Moskov. Mat. Obšč., 25, 119-262 (1971) · Zbl 0263.34003 |
| [7] | A. D. Brjuno, Analytic form of differential equations. II (Russian), Trudy Moskov. Mat. Obšč., 26, 199-239 (1972) · Zbl 0283.34013 |
| [8] | A. Bufetov; G. Forni, Limit theorems for horocycle flows, Ann. Sci. Éc. Norm. Supér., 47, 891-903 (2014) · Zbl 1360.37086 · doi:10.24033/asens.2229 |
| [9] | A. Bufetov, Limit theorems for translation flows, Ann. of Math., 179, 431-499 (2014) · Zbl 1290.37023 · doi:10.4007/annals.2014.179.2.2 |
| [10] | T. W. Cusick and M. E. Flahive, The Markoff and Lagrange spectra, in Mathematical Surverys and Monographs, 30, American Mathematical Society, 1989. · Zbl 0685.10023 |
| [11] | S. Marmi; D. H. Kim; L. Marchese, Long hitting time for translation flows and L-shaped billiards, J. Mod. Dyn., 14, 291-353 (2019) · Zbl 1442.37059 · doi:10.3934/jmd.2019011 |
| [12] | 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. · Zbl 1478.37002 |
| [13] | G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math., 146, 295-344 (1997) · Zbl 0893.58037 · doi:10.2307/2952464 |
| [14] | G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math., 155, 1-103 (2002) · Zbl 1034.37003 · doi:10.2307/3062150 |
| [15] | S. Ghazouani, Local rigidity for periodic generalised interval exchange transformations, Invent. Math., 226, 467-520 (2021) · Zbl 1489.37050 · doi:10.1007/s00222-021-01051-3 |
| [16] | S. Ghazouani and C. Ulcigrai, A priori bounds for giets, affine shadows and rigidity of foliations in genus 2, preprint, arXiv: 2106.03529, 2021. |
| [17] | M. Hall Jr., On the sum and product of continued fractions, Ann. of Math., 48, 966-993 (1947) · Zbl 0030.02201 · doi:10.2307/1969389 |
| [18] | M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math, 49, 5-233 (1979) · Zbl 0448.58019 |
| [19] | P. Hubert; S. Lelièvre; L. Marchese; C. Ulcigrai, The Lagrange spectrum of some square-tiled surfaces, Israel J. Math., 225, 553-607 (2018) · Zbl 1393.37045 · doi:10.1007/s11856-018-1667-3 |
| [20] | P. Hubert; L. Marchese; C. Ulcigrai, Lagrange spectra in Teichmüller dynamics via renormalization, Geom. Funct. Anal., 25, 180-255 (2015) · Zbl 1348.37051 · doi:10.1007/s00039-015-0321-z |
| [21] | M. Keane, Interval exchange transformations, Math. Z., 141, 25-31 (1975) · Zbl 0278.28010 · doi:10.1007/BF01236981 |
| [22] | M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26, 188-196 (1977) · Zbl 0351.28012 · doi:10.1007/BF03007668 |
| [23] | H. B. Keynes; D. Newton, A “minimal”, non-uniquely ergodic interval exchange transformation, Math. Z., 148, 101-105 (1976) · Zbl 0308.28014 · doi:10.1007/BF01214699 |
| [24] | D. H. Kim; S. Marmi, The recurrence time for interval exchange maps, Nonlinearity, 21, 2201-2210 (2008) · Zbl 1154.37341 · doi:10.1088/0951-7715/21/9/016 |
| [25] | 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. · Zbl 1484.11001 |
| [26] | 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. · Zbl 1134.37018 |
| [27] | S. Marmi; P. Moussa; J.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps, J. Amer. Math. Soc., 18, 823-872 (2005) · Zbl 1112.37002 · doi:10.1090/S0894-0347-05-00490-X |
| [28] | S. Marmi; P. Moussa; J.-C. Yoccoz, Affine interval exchange maps with a wandering interval, Proc. Lond. Math. Soc., 100, 639-669 (2010) · Zbl 1196.37041 · doi:10.1112/plms/pdp037 |
| [29] | S. Marmi; P. Moussa; J.-C. Yoccoz, Linearization of generalized interval exchange maps, Ann. of Math., 176, 1583-1646 (2012) · Zbl 1277.37067 · doi:10.4007/annals.2012.176.3.5 |
| [30] | S. Marmi; C. Ulcigrai; J.-C. Yoccoz, On Roth type conditions, duality and central Birkhoff sums for i.e.m., Astérisque, 416, 65-132 (2020) · Zbl 1467.37040 · doi:10.24033/ast |
| [31] | S. Marmi; J.-C. Yoccoz, Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps, Comm. Math. Phys., 344, 117-139 (2016) · Zbl 1360.37101 · doi:10.1007/s00220-016-2624-9 |
| [32] | H. Masur, Interval exchange transformations and measured foliations, Ann. of Math., 115, 169-200 (1982) · Zbl 0497.28012 · doi:10.2307/1971341 |
| [33] | K. F. Roth, Rational approximations to algebraic numbers, Mathematika, 2, 1-20 (1955) · Zbl 0064.28501 · doi:10.1112/S0025579300000644 |
| [34] | C. Series, The modular surface and continued fractions, J. London Math. Soc., 31, 69-80 (1985) · Zbl 0545.30001 · doi:10.1112/jlms/s2-31.1.69 |
| [35] | J. Smillie; B. Weiss, Minimal sets for flows on moduli space, Israel J. Math., 142, 249-260 (2004) · Zbl 1052.37025 · doi:10.1007/BF02771535 |
| [36] | W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., 115, 201-242 (1982) · Zbl 0486.28014 · doi:10.2307/1971391 |
| [37] | W. A. Veech, Boshernitzan’s criterion for unique ergodicity of an interval exchange transformation, Ergodic Theory Dynam. Systems, 7, 149-153 (1987) · Zbl 0657.28012 · doi:10.1017/S0143385700003862 |
| [38] | Y. B. Vorobets, Planar structures and billiards in rational polygons: The Veech alternative, Russian Math. Surveys, 51, 779-817 (1996) · Zbl 0897.58029 · doi:10.1070/RM1996v051n05ABEH002993 |
| [39] | 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. · Zbl 0595.57027 |
| [40] | J.-C. Yoccoz, Linéarisation des germes de difféomorphismes holomorphes de (C, 0), C. R. Acad. Sci. Paris Sér. I Math., 306, 55-58 (1988) · Zbl 0668.58010 |
| [41] | J.-C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, 231 (1995). · Zbl 0836.30001 |
| [42] | 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. · Zbl 1417.37153 |
| [43] | J.-C. Yoccoz, Continued fraction algorithms for interval exchange maps: An introduction, Frontiers in Number Theory, Physics and Geometry, 1, 401-435 (2006) · Zbl 1127.28011 |
| [44] | 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. · Zbl 1248.37038 |
| [45] | A. Zorich, Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17, 1477-1499 (1997) · Zbl 0958.37002 · doi:10.1017/S0143385797086215 |
| [46] | 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. · Zbl 0976.37012 |
| [47] | A. Zorich, Flat surfaces, Frontiers in Number Theory, Physics, and Geometry, 1, 437-583 (2006) · Zbl 1129.32012 |
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