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.

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[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.

[17]

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

[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.

[19]

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

[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.

[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.

[22]

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

[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).

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

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

[32]

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

[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.

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[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.

[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.

[43]

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

[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.

[45]

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

[46]

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

[47]

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

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.

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[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.

[17]

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

[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.

[19]

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

[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.

[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.

[22]

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

[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).

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

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

[32]

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

[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.

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[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.

[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.

[43]

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

[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.

[45]

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

[46]

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

[47]

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

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