February  2013, 33(2): 505-525. doi: 10.3934/dcds.2013.33.505

Expansive flows of surfaces

1. 

Departamento de Matemática y Estadística del Litoral, Universidad de la República, Gral. Rivera 1350, Salto, Uruguay

Received  August 2011 Revised  July 2012 Published  September 2012

We prove that a flow without singular points of index zero on a compact surface is expansive if and only if the singularities are of saddle type and the union of their separatrices is dense. Moreover we show that such flows are obtained by surgery on the suspension of minimal interval exchange maps.
Citation: Alfonso Artigue. Expansive flows of surfaces. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 505-525. doi: 10.3934/dcds.2013.33.505
References:
[1]

R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193. doi: 10.1016/0022-0396(72)90013-7.

[2]

M. Cobo, C. Gutiérrez and J. Llibre, Flows without wandering points on compact connected surfaces, Trans. Amer. Math. Soc., 362 (2010), 4569-4580. doi: 10.1090/S0002-9947-10-05113-5.

[3]

G. Gal'perin, T. Krüger and S. Troubetzkoy, Local instability of orbits in polygonal and polyhedral billiards, Comm. Math. Phys., 169 (1995), 463-473. doi: 10.1007/BF02099308.

[4]

C. Gutiérrez, Smoothability of Cherry flows on two-manifolds, In Lecture Notes in Math, Springer, Berlin, 1007 (1983), 308-331. doi: 10.1007/BFb0061422.

[5]

C. Gutiérrez, Smoothing continuous flows on two-manifolds and recurrences, Ergodic Theory Dynam. Systems, 6 (1986), 17-44.

[6]

P. Hartman, "Ordinary Differential Equations,'' John Wiley & Sons Inc., New York, 1964.

[7]

L. F. He and G. Z. Shan, The nonexistence of expansive flow on a compact 2-manifold, Chinese Ann. Math. Ser. B, 12 (1991), 213-218.

[8]

K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math, 27 (1990), 117-162.

[9]

M. W. Hirsch, "Differential Topology,'' Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York, 1976.

[10]

J. F. Jakobsen and W. R. Utz, The non-existence of expansive homeomorphisms on a closed 2-cell, Pacific J. Math, 10 (1960), 1319-1321.

[11]

M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.

[12]

M. Komuro, Expansive properties of Lorenz attractors, The theory of dynamical systems and its applications to nonlinear problems, (Kyoto, 1984), World Sci. Publishing, Singapore, 1984, 4-26.

[13]

J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Brasil. Mat. (N.S.), 20 (1989), 113-133. doi: 10.1007/BF02585472.

[14]

N. G. Markley, On the number of recurrent orbit closures, Proc. Amer. Math. Soc., 25 (1970), 413-416. doi: 10.1090/S0002-9939-1970-0256375-0.

[15]

A. Mayer, Trajectories on the closed orientable surfaces, Rec. Math. [Mat. Sbornik] N.S., 12 (1943), 71-84.

[16]

M. Oka, Expansiveness of real flows, Tsukuba J. Math, 14 (1990), 1-8.

[17]

H. Whitney, Regular families of curves, Ann. of Math, 34 (1933), 244-270. doi: 10.2307/1968202.

[18]

A. N. Zemljakov and A. B. Katok, Topological transitivity of billiards in polygons, Mat. Zametki, 18 (1975), 291-300.

show all references

References:
[1]

R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193. doi: 10.1016/0022-0396(72)90013-7.

[2]

M. Cobo, C. Gutiérrez and J. Llibre, Flows without wandering points on compact connected surfaces, Trans. Amer. Math. Soc., 362 (2010), 4569-4580. doi: 10.1090/S0002-9947-10-05113-5.

[3]

G. Gal'perin, T. Krüger and S. Troubetzkoy, Local instability of orbits in polygonal and polyhedral billiards, Comm. Math. Phys., 169 (1995), 463-473. doi: 10.1007/BF02099308.

[4]

C. Gutiérrez, Smoothability of Cherry flows on two-manifolds, In Lecture Notes in Math, Springer, Berlin, 1007 (1983), 308-331. doi: 10.1007/BFb0061422.

[5]

C. Gutiérrez, Smoothing continuous flows on two-manifolds and recurrences, Ergodic Theory Dynam. Systems, 6 (1986), 17-44.

[6]

P. Hartman, "Ordinary Differential Equations,'' John Wiley & Sons Inc., New York, 1964.

[7]

L. F. He and G. Z. Shan, The nonexistence of expansive flow on a compact 2-manifold, Chinese Ann. Math. Ser. B, 12 (1991), 213-218.

[8]

K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math, 27 (1990), 117-162.

[9]

M. W. Hirsch, "Differential Topology,'' Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York, 1976.

[10]

J. F. Jakobsen and W. R. Utz, The non-existence of expansive homeomorphisms on a closed 2-cell, Pacific J. Math, 10 (1960), 1319-1321.

[11]

M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.

[12]

M. Komuro, Expansive properties of Lorenz attractors, The theory of dynamical systems and its applications to nonlinear problems, (Kyoto, 1984), World Sci. Publishing, Singapore, 1984, 4-26.

[13]

J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Brasil. Mat. (N.S.), 20 (1989), 113-133. doi: 10.1007/BF02585472.

[14]

N. G. Markley, On the number of recurrent orbit closures, Proc. Amer. Math. Soc., 25 (1970), 413-416. doi: 10.1090/S0002-9939-1970-0256375-0.

[15]

A. Mayer, Trajectories on the closed orientable surfaces, Rec. Math. [Mat. Sbornik] N.S., 12 (1943), 71-84.

[16]

M. Oka, Expansiveness of real flows, Tsukuba J. Math, 14 (1990), 1-8.

[17]

H. Whitney, Regular families of curves, Ann. of Math, 34 (1933), 244-270. doi: 10.2307/1968202.

[18]

A. N. Zemljakov and A. B. Katok, Topological transitivity of billiards in polygons, Mat. Zametki, 18 (1975), 291-300.

[1]

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

[2]

Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35

[3]

Se-Hyun Ku. Expansive flows on uniform spaces. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1585-1598. doi: 10.3934/dcds.2021165

[4]

Alfonso Artigue. Singular cw-expansive flows. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2945-2956. doi: 10.3934/dcds.2017126

[5]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399

[6]

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

[7]

Woochul Jung, Ngocthach Nguyen, Yinong Yang. Spectral decomposition for rescaling expansive flows with rescaled shadowing. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2267-2283. doi: 10.3934/dcds.2020113

[8]

Tatsuya Arai. The structure of dendrites constructed by pointwise P-expansive maps on the unit interval. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 43-61. doi: 10.3934/dcds.2016.36.43

[9]

José Ginés Espín Buendía, Daniel Peralta-salas, Gabriel Soler López. Existence of minimal flows on nonorientable surfaces. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4191-4211. doi: 10.3934/dcds.2017178

[10]

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

[11]

Alfonso Artigue. Discrete and continuous topological dynamics: Fields of cross sections and expansive flows. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5911-5927. doi: 10.3934/dcds.2016059

[12]

Artur O. Lopes, Vladimir A. Rosas, Rafael O. Ruggiero. Cohomology and subcohomology problems for expansive, non Anosov geodesic flows. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 403-422. doi: 10.3934/dcds.2007.17.403

[13]

Dong Han Kim. The dynamical Borel-Cantelli lemma for interval maps. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 891-900. doi: 10.3934/dcds.2007.17.891

[14]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147

[15]

Dmitri Scheglov. Absence of mixing for smooth flows on genus two surfaces. Journal of Modern Dynamics, 2009, 3 (1) : 13-34. doi: 10.3934/jmd.2009.3.13

[16]

Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841

[17]

Dong Han Kim, Luca Marchese, Stefano Marmi. Long hitting time for translation flows and L-shaped billiards. Journal of Modern Dynamics, 2019, 14: 291-353. doi: 10.3934/jmd.2019011

[18]

Jacek Brzykcy, Krzysztof Frączek. Disjointness of interval exchange transformations from systems of probabilistic origin. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 53-73. doi: 10.3934/dcds.2010.27.53

[19]

Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013

[20]

David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (100)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]