American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A leading term for the velocity of stationary viscous incompressible flow around a rigid body performing a rotation and a translation
  • DCDS Home
  • This Issue
  • Next Article
    High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs
March  2017, 37(3): 1359-1387. doi: 10.3934/dcds.2017056

Discrete conley index theory for zero dimensional basic sets

Ketty A. De Rezende 1,, and  Mariana G. Villapouca 2,

1. 

Institute of Mathematics, Statistics and Scientific Computation, Universidade Estadual de Campinas, Campinas, SP 13.083-859, Brazil
2. 

Institute of Mathematics and Statistics, Universidade do Estado do Rio de Janeiro, Rio de Janeiro, RJ 20.550-900, Brazil

* Corresponding author

Received  May 2016 Revised  September 2016 Published  December 2016

Fund Project: The first author is partially supported by CNPq under grant 309734/2014-2 and by FAPESP under grant 2012/18780-0.

In this article the discrete Conley index theory is used to study diffeomorphisms on closed differentiable n-manifolds with zero dimensional hyperbolic chain recurrent set. A theorem is established for the computation of the discrete Conley index of these basic sets in terms of the dynamical information contained in their associated structure matrices. Also, a classification of the reduced homology Conley index of these basic sets is presented using its Jordan real form. This, in turn, is essential to obtain a characterization of a pair of connection matrices for a Morse decomposition of zero-dimensional basic sets of a diffeomorphism.

Keywords: Conley index, dynamical systems, homology theory, diffeomorphisms and connection matrix pair.
Mathematics Subject Classification: Primary:37B30, 37B10, 37C05;Secondary:37D15, 37D05, 37B35.
Citation: Ketty A. De Rezende, Mariana G. Villapouca. Discrete conley index theory for zero dimensional basic sets. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1359-1387. doi: 10.3934/dcds.2017056
References:
[1]

P. Bartiomiejczyk and Z. Dzedzej, Connection Matrix theory for discrete dynamical systems, Banach Center Publications, 47 (1999), 67-78.   Google Scholar

[2]

P. Blanchard and J. Franks, An obstruction to the existence of certain dynamics in surface diffeomorphisms, Ergodic Theory & Dynamical Systems, 1 (1981), 255-260.   Google Scholar

[3]

R. Bowen, Topological entropy and Axiom A, Proc. Sympos. Pure Math., 14 (1970), 23-41.   Google Scholar

[4]

R. Bowen and J. Franks, Homology for zero-dimensional nonwandering sets, Annals of Mathematics, 106 (1977), 73-92.  doi: 10.2307/1971159.  Google Scholar

[5]

C. Conley, Isolated Invariant Sets and Morse Index CBMS Regional Conference Series in Mathematics, n. 38, American Mathematical Society, Providence, R. I. , 1978.  Google Scholar

[6]

J. M. Franks, Homology and Dynamical Systems, CBMS Regional Conference Series in Mathematics, n. 49, American Mathematical Society, Providence, R. I. , 1982.  Google Scholar

[7]

J. M. Franks and D. S. Richeson, Shift equivalence and the Conley Index, Transactions of the American Mathematical Society, 352 (2000), 3305-3322.  doi: 10.1090/S0002-9947-00-02488-0.  Google Scholar

[8]

R. D. Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Transactions of the American Mathematical Society, 298 (1986), 193-213.  doi: 10.1090/S0002-9947-1986-0857439-7.  Google Scholar

[9]

R. D. Franzosa, The continuation theory for Morse decompositions and connection matrices, Transactions of the American Mathematical Society, 310 (1988), 781-803.  doi: 10.1090/S0002-9947-1988-0973177-6.  Google Scholar

[10]

R. D. Franzosa, The connection matrix theory for Morse decompositions, Transactions of the American Mathematical Society, 311 (1989), 561-592.  doi: 10.1090/S0002-9947-1989-0978368-7.  Google Scholar

[11]

M. Mrozek, Leray functor and cohomological Conley index for discrete dynamical systems, Transactions of the American Mathematical Society, 318 (1990), 149-178.  doi: 10.1090/S0002-9947-1990-0968888-1.  Google Scholar

[12]

J. F. Reineck, The connection matrix in Morse-Smale flows, Transactions of the American Mathematical Society, 322 (1990), 523-545.  doi: 10.1090/S0002-9947-1990-0972705-3.  Google Scholar

[13]

C. McCord and J. F. Reineck, Connection matrices and transition matrices, Banach Center Publications, 47 (1999), 41-55.   Google Scholar

[14]

D. S. Richeson, Connection Matrix Pairs for the Discrete Conley Index Ph. D thesis, Northwesten University, 1998.  Google Scholar

[15]

D. S. Richeson, Connection matrix pairs, Banach Center Publications, 47 (1999), 219-232.   Google Scholar

[16]

J. W. Robbin and D. Salamon, Dynamical systems, shape theory and Conley index, Ergodic Theory & Dynamical Systems, 8* (1988), 375{393. doi: 10.1017/S0143385700009494.  Google Scholar

[17]

D. Salamon, Connected simple systems and Conley index of isolated invariant sets, Transactions of the American Mathematical Society, 291 (1985), 1-41.  doi: 10.1090/S0002-9947-1985-0797044-3.  Google Scholar

[18]

D. Salamon, Morse Theory, Conley index and Floer homology, Bull. London Math. Soc., 22 (1990), 113-140.  doi: 10.1112/blms/22.2.113.  Google Scholar

[19]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 797-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[20]

M. Shub and D. Sullivan, Homology and dynamical systems, Topology, 14 (1975), 109-132.  doi: 10.1016/0040-9383(75)90022-1.  Google Scholar

[21]

A. Szymczak, The Conley index for discrete semidynamical systems, Topology and its Applications, 66 (1995), 215-240.  doi: 10.1016/0166-8641(95)0003J-S.  Google Scholar

show all references

References:
[1]

P. Bartiomiejczyk and Z. Dzedzej, Connection Matrix theory for discrete dynamical systems, Banach Center Publications, 47 (1999), 67-78.   Google Scholar

[2]

P. Blanchard and J. Franks, An obstruction to the existence of certain dynamics in surface diffeomorphisms, Ergodic Theory & Dynamical Systems, 1 (1981), 255-260.   Google Scholar

[3]

R. Bowen, Topological entropy and Axiom A, Proc. Sympos. Pure Math., 14 (1970), 23-41.   Google Scholar

[4]

R. Bowen and J. Franks, Homology for zero-dimensional nonwandering sets, Annals of Mathematics, 106 (1977), 73-92.  doi: 10.2307/1971159.  Google Scholar

[5]

C. Conley, Isolated Invariant Sets and Morse Index CBMS Regional Conference Series in Mathematics, n. 38, American Mathematical Society, Providence, R. I. , 1978.  Google Scholar

[6]

J. M. Franks, Homology and Dynamical Systems, CBMS Regional Conference Series in Mathematics, n. 49, American Mathematical Society, Providence, R. I. , 1982.  Google Scholar

[7]

J. M. Franks and D. S. Richeson, Shift equivalence and the Conley Index, Transactions of the American Mathematical Society, 352 (2000), 3305-3322.  doi: 10.1090/S0002-9947-00-02488-0.  Google Scholar

[8]

R. D. Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Transactions of the American Mathematical Society, 298 (1986), 193-213.  doi: 10.1090/S0002-9947-1986-0857439-7.  Google Scholar

[9]

R. D. Franzosa, The continuation theory for Morse decompositions and connection matrices, Transactions of the American Mathematical Society, 310 (1988), 781-803.  doi: 10.1090/S0002-9947-1988-0973177-6.  Google Scholar

[10]

R. D. Franzosa, The connection matrix theory for Morse decompositions, Transactions of the American Mathematical Society, 311 (1989), 561-592.  doi: 10.1090/S0002-9947-1989-0978368-7.  Google Scholar

[11]

M. Mrozek, Leray functor and cohomological Conley index for discrete dynamical systems, Transactions of the American Mathematical Society, 318 (1990), 149-178.  doi: 10.1090/S0002-9947-1990-0968888-1.  Google Scholar

[12]

J. F. Reineck, The connection matrix in Morse-Smale flows, Transactions of the American Mathematical Society, 322 (1990), 523-545.  doi: 10.1090/S0002-9947-1990-0972705-3.  Google Scholar

[13]

C. McCord and J. F. Reineck, Connection matrices and transition matrices, Banach Center Publications, 47 (1999), 41-55.   Google Scholar

[14]

D. S. Richeson, Connection Matrix Pairs for the Discrete Conley Index Ph. D thesis, Northwesten University, 1998.  Google Scholar

[15]

D. S. Richeson, Connection matrix pairs, Banach Center Publications, 47 (1999), 219-232.   Google Scholar

[16]

J. W. Robbin and D. Salamon, Dynamical systems, shape theory and Conley index, Ergodic Theory & Dynamical Systems, 8* (1988), 375{393. doi: 10.1017/S0143385700009494.  Google Scholar

[17]

D. Salamon, Connected simple systems and Conley index of isolated invariant sets, Transactions of the American Mathematical Society, 291 (1985), 1-41.  doi: 10.1090/S0002-9947-1985-0797044-3.  Google Scholar

[18]

D. Salamon, Morse Theory, Conley index and Floer homology, Bull. London Math. Soc., 22 (1990), 113-140.  doi: 10.1112/blms/22.2.113.  Google Scholar

[19]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 797-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[20]

M. Shub and D. Sullivan, Homology and dynamical systems, Topology, 14 (1975), 109-132.  doi: 10.1016/0040-9383(75)90022-1.  Google Scholar

[21]

A. Szymczak, The Conley index for discrete semidynamical systems, Topology and its Applications, 66 (1995), 215-240.  doi: 10.1016/0166-8641(95)0003J-S.  Google Scholar

Figure 1.  Braid diagram
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Graded module braids with endomorphisms 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Graded module braids with endomorphisms 2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Smale's Horseshoe
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  fitted diffeomorphism
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Smale diffeomorphism
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Connection matrix pair for zero-dimensional basic sets decomposition
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Representation of a connection matrix pair for a zero-dimensional basic set decomposition
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Fitted diffeomorphism
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387
[2]

Jintao Wang, Desheng Li, Jinqiao Duan. On the shape Conley index theory of semiflows on complete metric spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1629-1647. doi: 10.3934/dcds.2016.36.1629
[3]

Giovanni Russo, Fabian Wirth. Matrix measures, stability and contraction theory for dynamical systems on time scales. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021188
[4]

Robert Skiba, Nils Waterstraat. The index bundle and multiparameter bifurcation for discrete dynamical systems. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5603-5629. doi: 10.3934/dcds.2017243
[5]

M. C. Carbinatto, K. Mischaikow. Horseshoes and the Conley index spectrum - II: the theorem is sharp. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 599-616. doi: 10.3934/dcds.1999.5.599
[6]

Angelo B. Mingarelli. Nonlinear functionals in oscillation theory of matrix differential systems. Communications on Pure & Applied Analysis, 2004, 3 (1) : 75-84. doi: 10.3934/cpaa.2004.3.75
[7]

Kung-Ching Chang, Zhi-Qiang Wang, Tan Zhang. On a new index theory and non semi-trivial solutions for elliptic systems. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 809-826. doi: 10.3934/dcds.2010.28.809
[8]

Hahng-Yun Chu, Se-Hyun Ku, Jong-Suh Park. Conley's theorem for dispersive systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 313-321. doi: 10.3934/dcdss.2015.8.313
[9]

Marian Gidea. Leray functor and orbital Conley index for non-invariant sets. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 617-630. doi: 10.3934/dcds.1999.5.617
[10]

Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985
[11]

Carlos Matheus, Jean-Christophe Yoccoz. The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis. Journal of Modern Dynamics, 2010, 4 (3) : 453-486. doi: 10.3934/jmd.2010.4.453
[12]

Kazuyuki Yagasaki. Optimal control of the SIR epidemic model based on dynamical systems theory. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021144
[13]

Andrea Braides, Margherita Solci, Enrico Vitali. A derivation of linear elastic energies from pair-interaction atomistic systems. Networks & Heterogeneous Media, 2007, 2 (3) : 551-567. doi: 10.3934/nhm.2007.2.551
[14]

Simone Fiori. Auto-regressive moving-average discrete-time dynamical systems and autocorrelation functions on real-valued Riemannian matrix manifolds. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2785-2808. doi: 10.3934/dcdsb.2014.19.2785
[15]

Ugo Locatelli, Letizia Stefanelli. Quasi-periodic motions in a special class of dynamical equations with dissipative effects: A pair of detection methods. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1155-1187. doi: 10.3934/dcdsb.2015.20.1155
[16]

Michael Ghil. The wind-driven ocean circulation: Applying dynamical systems theory to a climate problem. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 189-228. doi: 10.3934/dcds.2017008
[17]

Rúben Sousa, Semyon Yakubovich. The spectral expansion approach to index transforms and connections with the theory of diffusion processes. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2351-2378. doi: 10.3934/cpaa.2018112
[18]

J. G. Ollason, N. Ren. A general dynamical theory of foraging in animals. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 713-720. doi: 10.3934/dcdsb.2004.4.713
[19]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
[20]

Michael Brin, Dmitri Burago, Sergey Ivanov. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. Journal of Modern Dynamics, 2009, 3 (1) : 1-11. doi: 10.3934/jmd.2009.3.1

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (108)
  • HTML views (54)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy