2022, 18: 131-147. doi: 10.3934/jmd.2022006

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

Received  November 21, 2021 Revised  February 06, 2022 Published  April 2022

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.

Citation: Stefano Marmi. Some arithmetical aspects of renormalization in Teichmüller dynamics: On the occasion of Corinna Ulcigrai winning the Brin Prize. Journal of Modern Dynamics, 2022, 18: 131-147. doi: 10.3934/jmd.2022006
References:
[1]

M. ArtigianiL. 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. ArtigianiL. 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. MarmiD. 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. FathiF. 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. HubertS. LelièvreL. 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. HubertL. 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. MarmiP. 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. MarmiP. 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. MarmiP. 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. MarmiC. 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. ArtigianiL. 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. ArtigianiL. 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. MarmiD. 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. FathiF. 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. HubertS. LelièvreL. 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. HubertL. 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. MarmiP. 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. MarmiP. 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. MarmiP. 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. MarmiC. 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. 

Figure 1.  The Brjuno function $ B(\alpha) $
Figure 2.  Rauzy diagram for 2 and 3 intervals
Figure 3.  Rauzy diagrams for 4 intervals and genus 2
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
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