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Ergodicity, mixing, Ratner's properties and disjointness for classical flows: On the research of Corinna Ulcigrai
Some arithmetical aspects of renormalization in Teichmüller dynamics: On the occasion of Corinna Ulcigrai winning the Brin Prize
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy |
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.
References:
[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.
doi: 10.4171/GGD/384. |
[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.
doi: 10.1017/etds.2018.143. |
[3] |
A. Avila and M. Viana,
Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56.
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 (1985), 723-752.
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.
doi: 10.19086/da.1749. |
[6] |
A. D. Brjuno,
Analytic form of differential equations. I (Russian), Trudy Moskov. Mat. Obšč., 25 (1971), 119-262.
|
[7] |
A. D. Brjuno,
Analytic form of differential equations. II (Russian), Trudy Moskov. Mat. Obšč., 26 (1972), 199-239.
|
[8] |
A. Bufetov and G. Forni,
Limit theorems for horocycle flows, Ann. Sci. Éc. Norm. Supér., 47 (2014), 891-903.
doi: 10.24033/asens.2229. |
[9] |
A. Bufetov,
Limit theorems for translation flows, Ann. of Math., 179 (2014), 431-499.
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.
doi: 10.1090/surv/030. |
[11] |
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.
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.
doi: 10.1007/s10240-018-0099-2. |
[13] |
A. Fathi, F. Laudenbach and V. Poénaru, Thurston's Work on Surfaces, Mathematical Notes, 48, Princeton University Press, Princeton, NJ, 2012.
![]() ![]() |
[14] |
G. Forni,
Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math., 146 (1997), 295-344.
doi: 10.2307/2952464. |
[15] |
G. Forni,
Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math., 155 (2002), 1-103.
doi: 10.2307/3062150. |
[16] |
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. |
[18] |
M. Hall Jr.,
On the sum and product of continued fractions, Ann. of Math., 48 (1947), 966-993.
doi: 10.2307/1969389. |
[19] |
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.
![]() ![]() |
[20] |
M.-R. Herman,
Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math, 49 (1979), 5-233.
|
[21] |
P. Hubert, S. Lelièvre, L. Marchese and C. Ulcigrai,
The Lagrange spectrum of some square-tiled surfaces, Israel J. Math., 225 (2018), 553-607.
doi: 10.1007/s11856-018-1667-3. |
[22] |
P. Hubert, L. Marchese and C. Ulcigrai,
Lagrange spectra in Teichmüller dynamics via renormalization, Geom. Funct. Anal., 25 (2015), 180-255.
doi: 10.1007/s00039-015-0321-z. |
[23] |
M. Keane,
Interval exchange transformations, Math. Z., 141 (1975), 25-31.
doi: 10.1007/BF01236981. |
[24] |
M. Keane,
Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196.
doi: 10.1007/BF03007668. |
[25] |
H. B. Keynes and D. Newton,
A "minimal", non-uniquely ergodic interval exchange transformation, Math. Z., 148 (1976), 101-105.
doi: 10.1007/BF01214699. |
[26] |
A. Y. Khinchin, Continued Fractions, University of Chicago Press, Chicago, Ill.-London, 1964.
![]() ![]() |
[27] |
D. H. Kim and S. Marmi,
The recurrence time for interval exchange maps, Nonlinearity, 21 (2008), 2201-2210.
doi: 10.1088/0951-7715/21/9/016. |
[28] |
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. |
[29] |
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. |
[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.
doi: 10.1090/S0894-0347-05-00490-X. |
[31] |
S. Marmi, P. Moussa and J.-C. Yoccoz,
Affine interval exchange maps with a wandering interval, Proc. Lond. Math. Soc., 100 (2010), 639-669.
doi: 10.1112/plms/pdp037. |
[32] |
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. |
[33] |
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.
doi: 10.24033/ast. |
[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. |
[35] |
H. Masur,
Interval exchange transformations and measured foliations, Ann. of Math., 115 (1982), 169-200.
doi: 10.2307/1971341. |
[36] |
K. F. Roth,
Rational approximations to algebraic numbers, Mathematika, 2 (1955), 1-20.
doi: 10.1112/S0025579300000644. |
[37] |
C. Series,
The modular surface and continued fractions, J. London Math. Soc., 31 (1985), 69-80.
doi: 10.1112/jlms/s2-31.1.69. |
[38] |
J. Smillie and B. Weiss,
Minimal sets for flows on moduli space, Israel J. Math., 142 (2004), 249-260.
doi: 10.1007/BF02771535. |
[39] |
W. A. Veech,
Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., 115 (1982), 201-242.
doi: 10.2307/1971391. |
[40] |
W. A. Veech,
Boshernitzan's criterion for unique ergodicity of an interval exchange transformation, Ergodic Theory Dynam. Systems, 7 (1987), 149-153.
doi: 10.1017/S0143385700003862. |
[41] |
Y. B. Vorobets,
Planar structures and billiards in rational polygons: The Veech alternative, Russian Math. Surveys, 51 (1996), 779-817.
doi: 10.1070/RM1996v051n05ABEH002993. |
[42] |
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.
doi: 10.24033/asens.1475. |
[43] |
J.-C. Yoccoz,
Linéarisation des germes de difféomorphismes holomorphes de (C, 0), C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 55-58.
|
[44] |
J.-C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, 231 (1995). |
[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.
|
[47] |
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. |
[48] |
A. Zorich,
Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477-1499.
doi: 10.1017/S0143385797086215. |
[49] |
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.
doi: 10.1090/trans2/197/05. |
[50] |
A. Zorich,
Flat surfaces, Frontiers in Number Theory, Physics, and Geometry, 1 (2006), 437-583.
|
show all references
References:
[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.
doi: 10.4171/GGD/384. |
[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.
doi: 10.1017/etds.2018.143. |
[3] |
A. Avila and M. Viana,
Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56.
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 (1985), 723-752.
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.
doi: 10.19086/da.1749. |
[6] |
A. D. Brjuno,
Analytic form of differential equations. I (Russian), Trudy Moskov. Mat. Obšč., 25 (1971), 119-262.
|
[7] |
A. D. Brjuno,
Analytic form of differential equations. II (Russian), Trudy Moskov. Mat. Obšč., 26 (1972), 199-239.
|
[8] |
A. Bufetov and G. Forni,
Limit theorems for horocycle flows, Ann. Sci. Éc. Norm. Supér., 47 (2014), 891-903.
doi: 10.24033/asens.2229. |
[9] |
A. Bufetov,
Limit theorems for translation flows, Ann. of Math., 179 (2014), 431-499.
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.
doi: 10.1090/surv/030. |
[11] |
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.
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.
doi: 10.1007/s10240-018-0099-2. |
[13] |
A. Fathi, F. Laudenbach and V. Poénaru, Thurston's Work on Surfaces, Mathematical Notes, 48, Princeton University Press, Princeton, NJ, 2012.
![]() ![]() |
[14] |
G. Forni,
Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math., 146 (1997), 295-344.
doi: 10.2307/2952464. |
[15] |
G. Forni,
Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math., 155 (2002), 1-103.
doi: 10.2307/3062150. |
[16] |
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. |
[18] |
M. Hall Jr.,
On the sum and product of continued fractions, Ann. of Math., 48 (1947), 966-993.
doi: 10.2307/1969389. |
[19] |
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.
![]() ![]() |
[20] |
M.-R. Herman,
Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math, 49 (1979), 5-233.
|
[21] |
P. Hubert, S. Lelièvre, L. Marchese and C. Ulcigrai,
The Lagrange spectrum of some square-tiled surfaces, Israel J. Math., 225 (2018), 553-607.
doi: 10.1007/s11856-018-1667-3. |
[22] |
P. Hubert, L. Marchese and C. Ulcigrai,
Lagrange spectra in Teichmüller dynamics via renormalization, Geom. Funct. Anal., 25 (2015), 180-255.
doi: 10.1007/s00039-015-0321-z. |
[23] |
M. Keane,
Interval exchange transformations, Math. Z., 141 (1975), 25-31.
doi: 10.1007/BF01236981. |
[24] |
M. Keane,
Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196.
doi: 10.1007/BF03007668. |
[25] |
H. B. Keynes and D. Newton,
A "minimal", non-uniquely ergodic interval exchange transformation, Math. Z., 148 (1976), 101-105.
doi: 10.1007/BF01214699. |
[26] |
A. Y. Khinchin, Continued Fractions, University of Chicago Press, Chicago, Ill.-London, 1964.
![]() ![]() |
[27] |
D. H. Kim and S. Marmi,
The recurrence time for interval exchange maps, Nonlinearity, 21 (2008), 2201-2210.
doi: 10.1088/0951-7715/21/9/016. |
[28] |
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. |
[29] |
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. |
[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.
doi: 10.1090/S0894-0347-05-00490-X. |
[31] |
S. Marmi, P. Moussa and J.-C. Yoccoz,
Affine interval exchange maps with a wandering interval, Proc. Lond. Math. Soc., 100 (2010), 639-669.
doi: 10.1112/plms/pdp037. |
[32] |
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. |
[33] |
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.
doi: 10.24033/ast. |
[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. |
[35] |
H. Masur,
Interval exchange transformations and measured foliations, Ann. of Math., 115 (1982), 169-200.
doi: 10.2307/1971341. |
[36] |
K. F. Roth,
Rational approximations to algebraic numbers, Mathematika, 2 (1955), 1-20.
doi: 10.1112/S0025579300000644. |
[37] |
C. Series,
The modular surface and continued fractions, J. London Math. Soc., 31 (1985), 69-80.
doi: 10.1112/jlms/s2-31.1.69. |
[38] |
J. Smillie and B. Weiss,
Minimal sets for flows on moduli space, Israel J. Math., 142 (2004), 249-260.
doi: 10.1007/BF02771535. |
[39] |
W. A. Veech,
Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., 115 (1982), 201-242.
doi: 10.2307/1971391. |
[40] |
W. A. Veech,
Boshernitzan's criterion for unique ergodicity of an interval exchange transformation, Ergodic Theory Dynam. Systems, 7 (1987), 149-153.
doi: 10.1017/S0143385700003862. |
[41] |
Y. B. Vorobets,
Planar structures and billiards in rational polygons: The Veech alternative, Russian Math. Surveys, 51 (1996), 779-817.
doi: 10.1070/RM1996v051n05ABEH002993. |
[42] |
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.
doi: 10.24033/asens.1475. |
[43] |
J.-C. Yoccoz,
Linéarisation des germes de difféomorphismes holomorphes de (C, 0), C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 55-58.
|
[44] |
J.-C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, 231 (1995). |
[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.
|
[47] |
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. |
[48] |
A. Zorich,
Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477-1499.
doi: 10.1017/S0143385797086215. |
[49] |
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.
doi: 10.1090/trans2/197/05. |
[50] |
A. Zorich,
Flat surfaces, Frontiers in Number Theory, Physics, and Geometry, 1 (2006), 437-583.
|



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