American Institute of Mathematical Sciences

Advanced
  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

Stefano Marmi

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.

Keywords: Diophantine conditions, Roth type, interval exchange maps, Birkhoff sums, deviation of ergodic averages, Lagrange spectrum, Veech surfaces, Hall ray.
Mathematics Subject Classification: Primary: 11J06; Secondary: 11J70, 37E05, 37E20.
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. 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. 

Figure 1.  The Brjuno function $ B(\alpha) $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Rauzy diagram for 2 and 3 intervals
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Rauzy diagrams for 4 intervals and genus 2
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Balázs Bárány, Michaƚ Rams, Ruxi Shi. On the multifractal spectrum of weighted Birkhoff averages. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2461-2497. doi: 10.3934/dcds.2021199
[2]

Andreas Strömbergsson. On the deviation of ergodic averages for horocycle flows. Journal of Modern Dynamics, 2013, 7 (2) : 291-328. doi: 10.3934/jmd.2013.7.291
[3]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
[4]

Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379
[5]

Jon Chaika, Howard Masur. There exists an interval exchange with a non-ergodic generic measure. Journal of Modern Dynamics, 2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289
[6]

Davit Karagulyan. Hausdorff dimension of a class of three-interval exchange maps. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1257-1281. doi: 10.3934/dcds.2020077
[7]

Eduard Duryev, Charles Fougeron, Selim Ghazouani. Dilation surfaces and their Veech groups. Journal of Modern Dynamics, 2019, 14: 121-151. doi: 10.3934/jmd.2019005
[8]

François Ledrappier, Omri Sarig. Fluctuations of ergodic sums for horocycle flows on $\Z^d$--covers of finite volume surfaces. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 247-325. doi: 10.3934/dcds.2008.22.247
[9]

Andreas Koutsogiannis. Multiple ergodic averages for tempered functions. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1177-1205. doi: 10.3934/dcds.2020314
[10]

Andreas Koutsogiannis. Multiple ergodic averages for variable polynomials. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022067
[11]

Xueting Tian. Topological pressure for the completely irregular set of birkhoff averages. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2745-2763. doi: 10.3934/dcds.2017118
[12]

Giovanni Forni, Carlos Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. Journal of Modern Dynamics, 2014, 8 (3&4) : 271-436. doi: 10.3934/jmd.2014.8.271
[13]

Andrew Best, Andreu Ferré Moragues. Polynomial ergodic averages for certain countable ring actions. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3379-3413. doi: 10.3934/dcds.2022019
[14]

Todd Young. Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 359-378. doi: 10.3934/dcds.2003.9.359
[15]

Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems and Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27
[16]

Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801
[17]

Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010
[18]

Pierre Arnoux, Thomas A. Schmidt. Veech surfaces with nonperiodic directions in the trace field. Journal of Modern Dynamics, 2009, 3 (4) : 611-629. doi: 10.3934/jmd.2009.3.611
[19]

Pascal Hubert, Gabriela Schmithüsen. Infinite translation surfaces with infinitely generated Veech groups. Journal of Modern Dynamics, 2010, 4 (4) : 715-732. doi: 10.3934/jmd.2010.4.715
[20]

Frédéric Naud. Birkhoff cones, symbolic dynamics and spectrum of transfer operators. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 581-598. doi: 10.3934/dcds.2004.11.581

2021 Impact Factor: 0.641

Tools

Metrics

  • PDF downloads (61)
  • HTML views (69)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy