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

Expansive flows of surfaces

Alfonso Artigue 1,

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.
Keywords: Dynamical systems, rational billiards., interval exchange maps, expansive flows, flows on surfaces.
Mathematics Subject Classification: Primary: 37E35; Secondary: 37E05, 37D5.
Citation: Alfonso Artigue. Expansive flows of surfaces. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

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