Expansive flows of surfaces

Previous Article
Inertial manifolds for a class of non-autonomous semilinear parabolic equations with finite delay
DCDS Home
This Issue
Next Article
Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices

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

Abstract
Related Papers

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 - A, 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. 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. 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. doi: 10.1007/BF02099308.
[4]	C. Gutiérrez, Smoothability of Cherry flows on two-manifolds,, In Lecture Notes in Math, 1007 (1983), 308. doi: 10.1007/BFb0061422.
[5]	C. Gutiérrez, Smoothing continuous flows on two-manifolds and recurrences,, Ergodic Theory Dynam. Systems, 6 (1986), 17.
[6]	P. Hartman, "Ordinary Differential Equations,'', John Wiley & Sons Inc., (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.
[8]	K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov,, Osaka J. Math, 27 (1990), 117.
[9]	M. W. Hirsch, "Differential Topology,'', Graduate Texts in Mathematics, (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.
[11]	M. Keane, Interval exchange transformations,, Math. Z., 141 (1975), 25. doi: 10.1007/BF01236981.
[12]	M. Komuro, Expansive properties of Lorenz attractors,, The theory of dynamical systems and its applications to nonlinear problems, (1984), 4.
[13]	J. Lewowicz, Expansive homeomorphisms of surfaces,, Bol. Soc. Brasil. Mat. (N.S.), 20 (1989), 113. doi: 10.1007/BF02585472.
[14]	N. G. Markley, On the number of recurrent orbit closures,, Proc. Amer. Math. Soc., 25 (1970), 413. 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.
[16]	M. Oka, Expansiveness of real flows,, Tsukuba J. Math, 14 (1990), 1.
[17]	H. Whitney, Regular families of curves,, Ann. of Math, 34 (1933), 244. doi: 10.2307/1968202.
[18]	A. N. Zemljakov and A. B. Katok, Topological transitivity of billiards in polygons,, Mat. Zametki, 18 (1975), 291.

show all references

References:

[1]	R. Bowen and P. Walters, Expansive one-parameter flows,, J. Differential Equations, 12 (1972), 180. 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. 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. doi: 10.1007/BF02099308.
[4]	C. Gutiérrez, Smoothability of Cherry flows on two-manifolds,, In Lecture Notes in Math, 1007 (1983), 308. doi: 10.1007/BFb0061422.
[5]	C. Gutiérrez, Smoothing continuous flows on two-manifolds and recurrences,, Ergodic Theory Dynam. Systems, 6 (1986), 17.
[6]	P. Hartman, "Ordinary Differential Equations,'', John Wiley & Sons Inc., (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.
[8]	K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov,, Osaka J. Math, 27 (1990), 117.
[9]	M. W. Hirsch, "Differential Topology,'', Graduate Texts in Mathematics, (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.
[11]	M. Keane, Interval exchange transformations,, Math. Z., 141 (1975), 25. doi: 10.1007/BF01236981.
[12]	M. Komuro, Expansive properties of Lorenz attractors,, The theory of dynamical systems and its applications to nonlinear problems, (1984), 4.
[13]	J. Lewowicz, Expansive homeomorphisms of surfaces,, Bol. Soc. Brasil. Mat. (N.S.), 20 (1989), 113. doi: 10.1007/BF02585472.
[14]	N. G. Markley, On the number of recurrent orbit closures,, Proc. Amer. Math. Soc., 25 (1970), 413. 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.
[16]	M. Oka, Expansiveness of real flows,, Tsukuba J. Math, 14 (1990), 1.
[17]	H. Whitney, Regular families of curves,, Ann. of Math, 34 (1933), 244. doi: 10.2307/1968202.
[18]	A. N. Zemljakov and A. B. Katok, Topological transitivity of billiards in polygons,, Mat. Zametki, 18 (1975), 291.

[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]	Alfonso Artigue. Singular cw-expansive flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2945-2956. doi: 10.3934/dcds.2017126
[4]	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
[5]	Tatsuya Arai. The structure of dendrites constructed by pointwise P-expansive maps on the unit interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 43-61. doi: 10.3934/dcds.2016.36.43
[6]	José Ginés Espín Buendía, Daniel Peralta-salas, Gabriel Soler López. Existence of minimal flows on nonorientable surfaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4191-4211. doi: 10.3934/dcds.2017178
[7]	Jacek Brzykcy, Krzysztof Frączek. Disjointness of interval exchange transformations from systems of probabilistic origin. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 53-73. doi: 10.3934/dcds.2010.27.53
[8]	Alfonso Artigue. Discrete and continuous topological dynamics: Fields of cross sections and expansive flows. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5911-5927. doi: 10.3934/dcds.2016059
[9]	Artur O. Lopes, Vladimir A. Rosas, Rafael O. Ruggiero. Cohomology and subcohomology problems for expansive, non Anosov geodesic flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 403-422. doi: 10.3934/dcds.2007.17.403
[10]	Dong Han Kim. The dynamical Borel-Cantelli lemma for interval maps. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 891-900. doi: 10.3934/dcds.2007.17.891
[11]	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
[12]	Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841
[13]	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
[14]	Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013
[15]	Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010
[16]	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
[17]	Alexander I. Bufetov. Hölder cocycles and ergodic integrals for translation flows on flat surfaces. Electronic Research Announcements, 2010, 17: 34-42. doi: 10.3934/era.2010.17.34
[18]	Giovanni Forni. The cohomological equation for area-preserving flows on compact surfaces. Electronic Research Announcements, 1995, 1: 114-123.
[19]	Matilde Martínez, Shigenori Matsumoto, Alberto Verjovsky. Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem. Journal of Modern Dynamics, 2016, 10: 113-134. doi: 10.3934/jmd.2016.10.113
[20]	Vladislav Kruglov, Dmitry Malyshev, Olga Pochinka. Topological classification of $Ω$-stable flows on surfaces by means of effectively distinguishable multigraphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4305-4327. doi: 10.3934/dcds.2018188

2017 Impact Factor: 1.179

American Institute of Mathematical Sciences