American Institute of Mathematical Sciences

Advanced
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

[1]

César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577
[2]

Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198
[3]

Daniel J. Thompson. A criterion for topological entropy to decrease under normalised Ricci flow. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1243-1248. doi: 10.3934/dcds.2011.30.1243
[4]

Gang Bao, Jun Lai. Radar cross section reduction of a cavity in the ground plane: TE polarization. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 419-434. doi: 10.3934/dcdss.2015.8.419
[5]

Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581
[6]

Ursula Hamenstädt. Dynamics of the Teichmüller flow on compact invariant sets. Journal of Modern Dynamics, 2010, 4 (2) : 393-418. doi: 10.3934/jmd.2010.4.393
[7]

Florent Berthelin, Paola Goatin. Regularity results for the solutions of a non-local model of traffic flow. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3197-3213. doi: 10.3934/dcds.2019132
[8]

Felisia Angela Chiarello, Paola Goatin. Non-local multi-class traffic flow models. Networks & Heterogeneous Media, 2019, 14 (2) : 371-387. doi: 10.3934/nhm.2019015
[9]

Nicolas Forcadel, Wilfredo Salazar, Mamdouh Zaydan. Homogenization of second order discrete model with local perturbation and application to traffic flow. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1437-1487. doi: 10.3934/dcds.2017060
[10]

Kien Ming Ng, Trung Hieu Tran. A parallel water flow algorithm with local search for solving the quadratic assignment problem. Journal of Industrial & Management Optimization, 2019, 15 (1) : 235-259. doi: 10.3934/jimo.2018041
[11]

Mohamed Benyahia, Massimiliano D. Rosini. A macroscopic traffic model with phase transitions and local point constraints on the flow. Networks & Heterogeneous Media, 2017, 12 (2) : 297-317. doi: 10.3934/nhm.2017013
[12]

Arnab Roy, Takéo Takahashi. Local null controllability of a rigid body moving into a Boussinesq flow. Mathematical Control & Related Fields, 2019, 9 (4) : 793-836. doi: 10.3934/mcrf.2019050
[13]

Zheng Sun, José A. Carrillo, Chi-Wang Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic & Related Models, 2019, 12 (4) : 885-908. doi: 10.3934/krm.2019033
[14]

Fabien Caubet, Carlos Conca, Matías Godoy. On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives. Inverse Problems & Imaging, 2016, 10 (2) : 327-367. doi: 10.3934/ipi.2016003
[15]

Anke D. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2173-2241. doi: 10.3934/dcds.2014.34.2173
[16]

Veronika Schleper. A hybrid model for traffic flow and crowd dynamics with random individual properties. Mathematical Biosciences & Engineering, 2015, 12 (2) : 393-413. doi: 10.3934/mbe.2015.12.393
[17]

Jun Li, Qi Wang. Flow driven dynamics of sheared flowing polymer-particulate nanocomposites. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1359-1382. doi: 10.3934/dcds.2010.26.1359
[18]

Dong Li, Tong Li. Shock formation in a traffic flow model with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2011, 6 (4) : 681-694. doi: 10.3934/nhm.2011.6.681
[19]

Joan Carles Tatjer, Arturo Vieiro. Dynamics of the QR-flow for upper Hessenberg real matrices. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1359-1403. doi: 10.3934/dcdsb.2020166
[20]

Mahdi Boukrouche, Grzegorz Łukaszewicz. On global in time dynamics of a planar Bingham flow subject to a subdifferential boundary condition. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 3969-3983. doi: 10.3934/dcds.2014.34.3969

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (124)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy