October  2016, 36(10): 5509-5553. doi: 10.3934/dcds.2016043

Infinite type flat surface models of ergodic systems

1. 

Department of Mathematics, University of Chicago, Chicago, Illinois 60637, United States

2. 

Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, United States

Received  March 2015 Revised  March 2016 Published  July 2016

We propose a general framework for constructing and describing infinite type flat surfaces of finite area. Using this method, we characterize the range of dynamical behaviors possible for the vertical translation flows on such flat surfaces. We prove a sufficient condition for ergodicity of this flow and apply the condition to several examples. We present specific examples of infinite type flat surfaces on which the translation flow exhibits dynamical phenomena not realizable by translation flows on finite type flat surfaces.
Citation: Kathryn Lindsey, Rodrigo Treviño. Infinite type flat surface models of ergodic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5509-5553. doi: 10.3934/dcds.2016043
References:
[1]

J. Aaronson and O. Sarig, Exponential chi-squared distributions in infinite ergodic theory,, Ergodic Theory and Dynamical Systems, 34 (2014), 705. doi: 10.1017/etds.2012.160. Google Scholar

[2]

T. M. Adams, Smorodinsky's conjecture on rank-one mixing,, Proc. Amer. Math. Soc., 126 (1998), 739. doi: 10.1090/S0002-9939-98-04082-9. Google Scholar

[3]

W. Ambrose, Representation of ergodic flows,, Ann. of Math. (2), 42 (1941), 723. doi: 10.2307/1969259. Google Scholar

[4]

P. Arnoux, D. S. Ornstein and B. Weiss, Cutting and stacking, interval exchanges and geometric models,, Israel J. Math., 50 (1985), 160. doi: 10.1007/BF02761122. Google Scholar

[5]

P. Arnoux and J.-C. Yoccoz, Construction de difféomorphismes pseudo-Anosov,, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 75. Google Scholar

[6]

S. Bezuglyi, J. Kwiatkowski and K. Medynets, Aperiodic substitution systems and their Bratteli diagrams,, Ergodic Theory Dynam. Systems, 29 (2009), 37. doi: 10.1017/S0143385708000230. Google Scholar

[7]

S. Bezuglyi, J. Kwiatkowski, K. Medynets and B. Solomyak, Invariant measures on stationary Bratteli diagrams,, Ergodic Theory Dynam. Systems, 30 (2010), 973. doi: 10.1017/S0143385709000443. Google Scholar

[8]

S. Bezuglyi, J. Kwiatkowski, K. Medynets and B. Solomyak, Finite rank Bratteli diagrams: Structure of invariant measures,, Trans. Amer. Math. Soc., 365 (2013), 2637. doi: 10.1090/S0002-9947-2012-05744-8. Google Scholar

[9]

R. Bowen and B. Marcus, Unique ergodicity for horocycle foliations,, Israel J. Math., 26 (1977), 43. doi: 10.1007/BF03007655. Google Scholar

[10]

J. Bowman, The complete family of Arnoux-Yoccoz surfaces,, Geometriae Dedicata, 164 (2013), 113. doi: 10.1007/s10711-012-9762-9. Google Scholar

[11]

O. Bratteli, Inductive limits of finite dimensional $C^{*} $-algebras,, Trans. Amer. Math. Soc., 171 (1972), 195. Google Scholar

[12]

A. I. Bufetov, Limit theorems for suspension flows over vershik automorphisms,, Russian Mathematical Surveys, 68 (2013), 789. Google Scholar

[13]

A. I. Bufetov, Finitely-additive measures on the asymptotic foliations of a Markov compactum,, Mosc. Math. J., 14 (2014), 205. Google Scholar

[14]

A. I. Bufetov, Limit theorems for translation flows,, Ann. of Math. (2), 179 (2014), 431. doi: 10.4007/annals.2014.179.2.2. Google Scholar

[15]

R. V. Chacon, Weakly mixing transformations which are not strongly mixing,, Proc. Amer. Math. Soc., 22 (1969), 559. doi: 10.1090/S0002-9939-1969-0247028-5. Google Scholar

[16]

R. Chamanara, Affine automorphism groups of surfaces of infinite type,, In In the tradition of Ahlfors and Bers, (2004), 123. doi: 10.1090/conm/355/06449. Google Scholar

[17]

D. Creutz and C. E. Silva, Mixing on rank-one transformations,, Studia Math., 199 (2010), 43. doi: 10.4064/sm199-1-4. Google Scholar

[18]

M. D. Esposti, G. Del Magno and M. Lenci, Escape orbits and ergodicity in infinite step billiards,, Nonlinearity, 13 (2000), 1275. doi: 10.1088/0951-7715/13/4/316. Google Scholar

[19]

V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model,, Ann. Sci. Éc. Norm. Supér., 47 (2014), 1085. Google Scholar

[20]

F. Durand, B. Host and C. Skau, Substitutional dynamical systems, Bratteli diagrams and dimension groups,, Ergodic Theory Dynam. Systems, 19 (1999), 953. doi: 10.1017/S0143385799133947. Google Scholar

[21]

S. Ferenczi, A. M. Fisher and M. Talet, Minimality and unique ergodicity for adic transformations,, J. Anal. Math., 109 (2009), 1. doi: 10.1007/s11854-009-0027-y. Google Scholar

[22]

A. M. Fisher, Nonstationary mixing and the unique ergodicity of adic transformations,, Stoch. Dyn., 9 (2009), 335. doi: 10.1142/S0219493709002701. Google Scholar

[23]

G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards,, ArXiv e-prints, (2013). Google Scholar

[24]

K. Frączek and C. Ulcigrai, Non-ergodic Z-periodic billiards and infinite translation surfaces,, Invent. Math., 197 (2014), 241. doi: 10.1007/s00222-013-0482-z. Google Scholar

[25]

R. Gjerde and Ø. Johansen, Bratteli-Vershik models for Cantor minimal systems associated to interval exchange transformations,, Math. Scand., 90 (2002), 87. Google Scholar

[26]

R. H. Herman, I. F. Putnam and C. F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics,, Internat. J. Math., 3 (1992), 827. doi: 10.1142/S0129167X92000382. Google Scholar

[27]

W. Patrick Hooper, The invariant measures of some infinite interval exchange maps,, Geom. Topol., 19 (2015), 1895. doi: 10.2140/gt.2015.19.1895. Google Scholar

[28]

W. Patrick Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase,, Discrete Contin. Dyn. Syst., 33 (2013), 4341. doi: 10.3934/dcds.2013.33.4341. Google Scholar

[29]

P. Hubert and E. Lanneau, Veech groups without parabolic elements,, Duke Math. J., 133 (2006), 335. doi: 10.1215/S0012-7094-06-13326-4. Google Scholar

[30]

P. Hubert, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion,, J. Reine Angew. Math., 656 (2011), 223. doi: 10.1515/CRELLE.2011.052. Google Scholar

[31]

A. Katok, Interval exchange transformations and some special flows are not mixing,, Israel J. Math., 35 (1980), 301. doi: 10.1007/BF02760655. Google Scholar

[32]

X. Méla and K. Petersen, Dynamical properties of the Pascal adic transformation,, Ergodic Theory Dynam. Systems, 25 (2005), 227. doi: 10.1017/S0143385704000173. Google Scholar

[33]

D. S. Ornstein, On the root problem in ergodic theory,, In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, (1970), 347. Google Scholar

[34]

K. Petersen and K. Schmidt, Symmetric Gibbs measures,, Trans. Amer. Math. Soc., 349 (1997), 2775. doi: 10.1090/S0002-9947-97-01934-X. Google Scholar

[35]

D. Ralston and S. Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces,, J. Mod. Dyn., 6 (2012), 477. Google Scholar

[36]

G. Rauzy, Une généralisation du développement en fraction continue,, In Séminaire Delange-Pisot-Poitou, (1976). Google Scholar

[37]

G. Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315. Google Scholar

[38]

D. Rudolph, A two-valued step coding for ergodic flows,, Math. Z., 150 (1976), 201. doi: 10.1007/BF01221147. Google Scholar

[39]

P. Shields, Cutting and independent stacking of intervals,, Math. Systems Theory, 7 (1973), 1. doi: 10.1007/BF01824799. Google Scholar

[40]

C. E. Silva, Invitation to Ergodic Theory, volume 42 of Student Mathematical Library,, American Mathematical Society, (2008). Google Scholar

[41]

K. Strebel, Quadratic Differentials, volume 5 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)],, Springer-Verlag, (1984). doi: 10.1007/978-3-662-02414-0. Google Scholar

[42]

R. Treviño, On the ergodicity of flat surfaces of finite area,, Geom. Funct. Anal., 24 (2014), 360. doi: 10.1007/s00039-014-0269-4. Google Scholar

[43]

W. A. Veech, Interval exchange transformations,, J. Analyse Math., 33 (1978), 222. doi: 10.1007/BF02790174. Google Scholar

[44]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391. Google Scholar

[45]

A. M. Vershik, A new model of the ergodic transformations,, In Dynamical systems and ergodic theory (Warsaw, (1986), 381. Google Scholar

[46]

M. Viana, Dynamics of Interval Exchange Transformations and Teichmüller Flows,, Lecture Notes, (2008). Google Scholar

[47]

A. Zorich, Flat surfaces,, In Frontiers in number theory, (2006), 437. doi: 10.1007/978-3-540-31347-2_13. Google Scholar

show all references

References:
[1]

J. Aaronson and O. Sarig, Exponential chi-squared distributions in infinite ergodic theory,, Ergodic Theory and Dynamical Systems, 34 (2014), 705. doi: 10.1017/etds.2012.160. Google Scholar

[2]

T. M. Adams, Smorodinsky's conjecture on rank-one mixing,, Proc. Amer. Math. Soc., 126 (1998), 739. doi: 10.1090/S0002-9939-98-04082-9. Google Scholar

[3]

W. Ambrose, Representation of ergodic flows,, Ann. of Math. (2), 42 (1941), 723. doi: 10.2307/1969259. Google Scholar

[4]

P. Arnoux, D. S. Ornstein and B. Weiss, Cutting and stacking, interval exchanges and geometric models,, Israel J. Math., 50 (1985), 160. doi: 10.1007/BF02761122. Google Scholar

[5]

P. Arnoux and J.-C. Yoccoz, Construction de difféomorphismes pseudo-Anosov,, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 75. Google Scholar

[6]

S. Bezuglyi, J. Kwiatkowski and K. Medynets, Aperiodic substitution systems and their Bratteli diagrams,, Ergodic Theory Dynam. Systems, 29 (2009), 37. doi: 10.1017/S0143385708000230. Google Scholar

[7]

S. Bezuglyi, J. Kwiatkowski, K. Medynets and B. Solomyak, Invariant measures on stationary Bratteli diagrams,, Ergodic Theory Dynam. Systems, 30 (2010), 973. doi: 10.1017/S0143385709000443. Google Scholar

[8]

S. Bezuglyi, J. Kwiatkowski, K. Medynets and B. Solomyak, Finite rank Bratteli diagrams: Structure of invariant measures,, Trans. Amer. Math. Soc., 365 (2013), 2637. doi: 10.1090/S0002-9947-2012-05744-8. Google Scholar

[9]

R. Bowen and B. Marcus, Unique ergodicity for horocycle foliations,, Israel J. Math., 26 (1977), 43. doi: 10.1007/BF03007655. Google Scholar

[10]

J. Bowman, The complete family of Arnoux-Yoccoz surfaces,, Geometriae Dedicata, 164 (2013), 113. doi: 10.1007/s10711-012-9762-9. Google Scholar

[11]

O. Bratteli, Inductive limits of finite dimensional $C^{*} $-algebras,, Trans. Amer. Math. Soc., 171 (1972), 195. Google Scholar

[12]

A. I. Bufetov, Limit theorems for suspension flows over vershik automorphisms,, Russian Mathematical Surveys, 68 (2013), 789. Google Scholar

[13]

A. I. Bufetov, Finitely-additive measures on the asymptotic foliations of a Markov compactum,, Mosc. Math. J., 14 (2014), 205. Google Scholar

[14]

A. I. Bufetov, Limit theorems for translation flows,, Ann. of Math. (2), 179 (2014), 431. doi: 10.4007/annals.2014.179.2.2. Google Scholar

[15]

R. V. Chacon, Weakly mixing transformations which are not strongly mixing,, Proc. Amer. Math. Soc., 22 (1969), 559. doi: 10.1090/S0002-9939-1969-0247028-5. Google Scholar

[16]

R. Chamanara, Affine automorphism groups of surfaces of infinite type,, In In the tradition of Ahlfors and Bers, (2004), 123. doi: 10.1090/conm/355/06449. Google Scholar

[17]

D. Creutz and C. E. Silva, Mixing on rank-one transformations,, Studia Math., 199 (2010), 43. doi: 10.4064/sm199-1-4. Google Scholar

[18]

M. D. Esposti, G. Del Magno and M. Lenci, Escape orbits and ergodicity in infinite step billiards,, Nonlinearity, 13 (2000), 1275. doi: 10.1088/0951-7715/13/4/316. Google Scholar

[19]

V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model,, Ann. Sci. Éc. Norm. Supér., 47 (2014), 1085. Google Scholar

[20]

F. Durand, B. Host and C. Skau, Substitutional dynamical systems, Bratteli diagrams and dimension groups,, Ergodic Theory Dynam. Systems, 19 (1999), 953. doi: 10.1017/S0143385799133947. Google Scholar

[21]

S. Ferenczi, A. M. Fisher and M. Talet, Minimality and unique ergodicity for adic transformations,, J. Anal. Math., 109 (2009), 1. doi: 10.1007/s11854-009-0027-y. Google Scholar

[22]

A. M. Fisher, Nonstationary mixing and the unique ergodicity of adic transformations,, Stoch. Dyn., 9 (2009), 335. doi: 10.1142/S0219493709002701. Google Scholar

[23]

G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards,, ArXiv e-prints, (2013). Google Scholar

[24]

K. Frączek and C. Ulcigrai, Non-ergodic Z-periodic billiards and infinite translation surfaces,, Invent. Math., 197 (2014), 241. doi: 10.1007/s00222-013-0482-z. Google Scholar

[25]

R. Gjerde and Ø. Johansen, Bratteli-Vershik models for Cantor minimal systems associated to interval exchange transformations,, Math. Scand., 90 (2002), 87. Google Scholar

[26]

R. H. Herman, I. F. Putnam and C. F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics,, Internat. J. Math., 3 (1992), 827. doi: 10.1142/S0129167X92000382. Google Scholar

[27]

W. Patrick Hooper, The invariant measures of some infinite interval exchange maps,, Geom. Topol., 19 (2015), 1895. doi: 10.2140/gt.2015.19.1895. Google Scholar

[28]

W. Patrick Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase,, Discrete Contin. Dyn. Syst., 33 (2013), 4341. doi: 10.3934/dcds.2013.33.4341. Google Scholar

[29]

P. Hubert and E. Lanneau, Veech groups without parabolic elements,, Duke Math. J., 133 (2006), 335. doi: 10.1215/S0012-7094-06-13326-4. Google Scholar

[30]

P. Hubert, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion,, J. Reine Angew. Math., 656 (2011), 223. doi: 10.1515/CRELLE.2011.052. Google Scholar

[31]

A. Katok, Interval exchange transformations and some special flows are not mixing,, Israel J. Math., 35 (1980), 301. doi: 10.1007/BF02760655. Google Scholar

[32]

X. Méla and K. Petersen, Dynamical properties of the Pascal adic transformation,, Ergodic Theory Dynam. Systems, 25 (2005), 227. doi: 10.1017/S0143385704000173. Google Scholar

[33]

D. S. Ornstein, On the root problem in ergodic theory,, In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, (1970), 347. Google Scholar

[34]

K. Petersen and K. Schmidt, Symmetric Gibbs measures,, Trans. Amer. Math. Soc., 349 (1997), 2775. doi: 10.1090/S0002-9947-97-01934-X. Google Scholar

[35]

D. Ralston and S. Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces,, J. Mod. Dyn., 6 (2012), 477. Google Scholar

[36]

G. Rauzy, Une généralisation du développement en fraction continue,, In Séminaire Delange-Pisot-Poitou, (1976). Google Scholar

[37]

G. Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315. Google Scholar

[38]

D. Rudolph, A two-valued step coding for ergodic flows,, Math. Z., 150 (1976), 201. doi: 10.1007/BF01221147. Google Scholar

[39]

P. Shields, Cutting and independent stacking of intervals,, Math. Systems Theory, 7 (1973), 1. doi: 10.1007/BF01824799. Google Scholar

[40]

C. E. Silva, Invitation to Ergodic Theory, volume 42 of Student Mathematical Library,, American Mathematical Society, (2008). Google Scholar

[41]

K. Strebel, Quadratic Differentials, volume 5 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)],, Springer-Verlag, (1984). doi: 10.1007/978-3-662-02414-0. Google Scholar

[42]

R. Treviño, On the ergodicity of flat surfaces of finite area,, Geom. Funct. Anal., 24 (2014), 360. doi: 10.1007/s00039-014-0269-4. Google Scholar

[43]

W. A. Veech, Interval exchange transformations,, J. Analyse Math., 33 (1978), 222. doi: 10.1007/BF02790174. Google Scholar

[44]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391. Google Scholar

[45]

A. M. Vershik, A new model of the ergodic transformations,, In Dynamical systems and ergodic theory (Warsaw, (1986), 381. Google Scholar

[46]

M. Viana, Dynamics of Interval Exchange Transformations and Teichmüller Flows,, Lecture Notes, (2008). Google Scholar

[47]

A. Zorich, Flat surfaces,, In Frontiers in number theory, (2006), 437. doi: 10.1007/978-3-540-31347-2_13. Google Scholar

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