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November  2016, 36(11): 5911-5927. doi: 10.3934/dcds.2016059

Discrete and continuous topological dynamics: Fields of cross sections and expansive flows

Alfonso Artigue 1,

1. 

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

Received  February 2015 Revised  May 2016 Published  August 2016

In this article we consider the general problem of translating definitions and results from the category of discrete-time dynamical systems to the category of flows. We consider the dynamics of homeomorphisms and flows on compact metric spaces, in particular Peano continua. As a translating tool, we construct continuous, symmetric and monotonous fields of local cross sections for an arbitrary flow without singular points. Next, we use this structure in the study of expansive flows on Peano continua. We show that expansive flows have not stable points and that every point contains a non-trivial continuum in its stable set. As a corollary we obtain that no Peano continuum with an open set homeomorphic to the plane admits an expansive flow. In particular, compact surfaces do not admit expansive flows without singular points.
Keywords: Topological dynamics, local cross section, flow, homeomorphism, expansive flow..
Mathematics Subject Classification: 37B45, 37B0.
Citation: Alfonso Artigue. Discrete and continuous topological dynamics: Fields of cross sections and expansive flows. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5911-5927. doi: 10.3934/dcds.2016059
References:
[1]

A. Artigue, Positive expansive flows, Topology Appl., 165 (2014), 121-132. doi: 10.1016/j.topol.2014.01.015.  Google Scholar

[2]

R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc., 55 (1949), 1101-1110. doi: 10.1090/S0002-9904-1949-09334-5.  Google Scholar

[3]

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

[4]

L. W. Flinn, Expansive Flows, Phd Thesis, University of Warwick, 1972. Google Scholar

[5]

S. Goodman, Vector fields with transverse foliations, Topology, 24 (1985), 333-340. doi: 10.1016/0040-9383(85)90005-9.  Google Scholar

[6]

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

[7]

A. Illanes and S. B. Nadler Jr., Hyperspaces: Fundamentals and Recent Advances, Marcel Dekker, Inc., 1999.  Google Scholar

[8]

K. Kawamura, A direct proof that each Peano continuum with a free arc admits no expansive homeomorphism, Tsukuba J. Math., 12 (1988), 521-524.  Google Scholar

[9]

H. B. Keynes and M. Sears, Real-expansive flows and topological dimension, Ergodic Theory Dynam. Systems, 1 (1981), 179-195.  Google Scholar

[10]

J. Lewowicz, Lyapunov functions and stability of geodesic flows, Lecture Notes in Math., 1007 (1983), Springer, 463-479. doi: 10.1007/BFb0061429.  Google Scholar

[11]

J. Milnor, Microbundles Part I, Topology, 3 (1964), 53-80. doi: 10.1016/0040-9383(64)90005-9.  Google Scholar

[12]

E. E. Moise, Grille decomposition and convexification theorems for compact metric locally connected continua, Bull. Amer. Math. Soc., 55 (1949), 1111-1121. doi: 10.1090/S0002-9904-1949-09336-9.  Google Scholar

[13]

K. Moriyasu, K. Sakai and W. Sun, $C^1$-stably expansive flows, J. Differential Equations, 213 (2005), 352-367. doi: 10.1016/j.jde.2004.08.003.  Google Scholar

[14]

M. Oka, Singular foliations on cross-sections of expansive flows on 3-manifolds, Osaka J. Math., 27 (1990), 863-883.  Google Scholar

[15]

M. Paternain, Expansive flows and the fundamental group, Bull. Braz. Math. Soc., 24 (1993), 179-199. doi: 10.1007/BF01237676.  Google Scholar

[16]

R. F. Thomas, Entropy of expansive flows, Ergodic Theory Dynam. Systems, 7 (1987), 611-625. doi: 10.1017/S0143385700004235.  Google Scholar

[17]

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

show all references

References:
[1]

A. Artigue, Positive expansive flows, Topology Appl., 165 (2014), 121-132. doi: 10.1016/j.topol.2014.01.015.  Google Scholar

[2]

R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc., 55 (1949), 1101-1110. doi: 10.1090/S0002-9904-1949-09334-5.  Google Scholar

[3]

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

[4]

L. W. Flinn, Expansive Flows, Phd Thesis, University of Warwick, 1972. Google Scholar

[5]

S. Goodman, Vector fields with transverse foliations, Topology, 24 (1985), 333-340. doi: 10.1016/0040-9383(85)90005-9.  Google Scholar

[6]

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

[7]

A. Illanes and S. B. Nadler Jr., Hyperspaces: Fundamentals and Recent Advances, Marcel Dekker, Inc., 1999.  Google Scholar

[8]

K. Kawamura, A direct proof that each Peano continuum with a free arc admits no expansive homeomorphism, Tsukuba J. Math., 12 (1988), 521-524.  Google Scholar

[9]

H. B. Keynes and M. Sears, Real-expansive flows and topological dimension, Ergodic Theory Dynam. Systems, 1 (1981), 179-195.  Google Scholar

[10]

J. Lewowicz, Lyapunov functions and stability of geodesic flows, Lecture Notes in Math., 1007 (1983), Springer, 463-479. doi: 10.1007/BFb0061429.  Google Scholar

[11]

J. Milnor, Microbundles Part I, Topology, 3 (1964), 53-80. doi: 10.1016/0040-9383(64)90005-9.  Google Scholar

[12]

E. E. Moise, Grille decomposition and convexification theorems for compact metric locally connected continua, Bull. Amer. Math. Soc., 55 (1949), 1111-1121. doi: 10.1090/S0002-9904-1949-09336-9.  Google Scholar

[13]

K. Moriyasu, K. Sakai and W. Sun, $C^1$-stably expansive flows, J. Differential Equations, 213 (2005), 352-367. doi: 10.1016/j.jde.2004.08.003.  Google Scholar

[14]

M. Oka, Singular foliations on cross-sections of expansive flows on 3-manifolds, Osaka J. Math., 27 (1990), 863-883.  Google Scholar

[15]

M. Paternain, Expansive flows and the fundamental group, Bull. Braz. Math. Soc., 24 (1993), 179-199. doi: 10.1007/BF01237676.  Google Scholar

[16]

R. F. Thomas, Entropy of expansive flows, Ergodic Theory Dynam. Systems, 7 (1987), 611-625. doi: 10.1017/S0143385700004235.  Google Scholar

[17]

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

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