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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 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.

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