# American Institute of Mathematical Sciences

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

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

 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.
Citation: 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
##### References:
 [1] A. Artigue, Positive expansive flows, Topology Appl., 165 (2014), 121-132. doi: 10.1016/j.topol.2014.01.015. [2] R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc., 55 (1949), 1101-1110. doi: 10.1090/S0002-9904-1949-09334-5. [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. [4] L. W. Flinn, Expansive Flows, Phd Thesis, University of Warwick, 1972. [5] S. Goodman, Vector fields with transverse foliations, Topology, 24 (1985), 333-340. doi: 10.1016/0040-9383(85)90005-9. [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. [7] A. Illanes and S. B. Nadler Jr., Hyperspaces: Fundamentals and Recent Advances, Marcel Dekker, Inc., 1999. [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. [9] H. B. Keynes and M. Sears, Real-expansive flows and topological dimension, Ergodic Theory Dynam. Systems, 1 (1981), 179-195. [10] J. Lewowicz, Lyapunov functions and stability of geodesic flows, Lecture Notes in Math., 1007 (1983), Springer, 463-479. doi: 10.1007/BFb0061429. [11] J. Milnor, Microbundles Part I, Topology, 3 (1964), 53-80. doi: 10.1016/0040-9383(64)90005-9. [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. [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. [14] M. Oka, Singular foliations on cross-sections of expansive flows on 3-manifolds, Osaka J. Math., 27 (1990), 863-883. [15] M. Paternain, Expansive flows and the fundamental group, Bull. Braz. Math. Soc., 24 (1993), 179-199. doi: 10.1007/BF01237676. [16] R. F. Thomas, Entropy of expansive flows, Ergodic Theory Dynam. Systems, 7 (1987), 611-625. doi: 10.1017/S0143385700004235. [17] H. Whitney, Regular family of curves, Ann. of Math., 34 (1933), 244-270. doi: 10.2307/1968202.

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##### References:
 [1] A. Artigue, Positive expansive flows, Topology Appl., 165 (2014), 121-132. doi: 10.1016/j.topol.2014.01.015. [2] R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc., 55 (1949), 1101-1110. doi: 10.1090/S0002-9904-1949-09334-5. [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. [4] L. W. Flinn, Expansive Flows, Phd Thesis, University of Warwick, 1972. [5] S. Goodman, Vector fields with transverse foliations, Topology, 24 (1985), 333-340. doi: 10.1016/0040-9383(85)90005-9. [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. [7] A. Illanes and S. B. Nadler Jr., Hyperspaces: Fundamentals and Recent Advances, Marcel Dekker, Inc., 1999. [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. [9] H. B. Keynes and M. Sears, Real-expansive flows and topological dimension, Ergodic Theory Dynam. Systems, 1 (1981), 179-195. [10] J. Lewowicz, Lyapunov functions and stability of geodesic flows, Lecture Notes in Math., 1007 (1983), Springer, 463-479. doi: 10.1007/BFb0061429. [11] J. Milnor, Microbundles Part I, Topology, 3 (1964), 53-80. doi: 10.1016/0040-9383(64)90005-9. [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. [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. [14] M. Oka, Singular foliations on cross-sections of expansive flows on 3-manifolds, Osaka J. Math., 27 (1990), 863-883. [15] M. Paternain, Expansive flows and the fundamental group, Bull. Braz. Math. Soc., 24 (1993), 179-199. doi: 10.1007/BF01237676. [16] R. F. Thomas, Entropy of expansive flows, Ergodic Theory Dynam. Systems, 7 (1987), 611-625. doi: 10.1017/S0143385700004235. [17] H. Whitney, Regular family of curves, Ann. of Math., 34 (1933), 244-270. doi: 10.2307/1968202.
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