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2017, 37(3): 1359-1387.
doi: 10.3934/dcds.2017056
Discrete conley index theory for zero dimensional basic sets
| 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 |
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 and 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.
|
| [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.
|
| [3] |
R. Bowen,
Topological entropy and Axiom A, Proc. Sympos. Pure Math., 14 (1970), 23-41.
|
| [4] |
R. Bowen and J. Franks,
Homology for zero-dimensional nonwandering sets, Annals of Mathematics, 106 (1977), 73-92.
doi: 10.2307/1971159. |
| [5] |
C. Conley,
Isolated Invariant Sets and Morse Index CBMS Regional Conference Series in Mathematics, n. 38, American Mathematical Society, Providence, R. I. , 1978. |
| [6] |
J. M. Franks,
Homology and Dynamical Systems, CBMS Regional Conference Series in Mathematics, n. 49, American Mathematical Society, Providence, R. I. , 1982. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [13] |
C. McCord and J. F. Reineck,
Connection matrices and transition matrices, Banach Center Publications, 47 (1999), 41-55.
|
| [14] |
D. S. Richeson,
Connection Matrix Pairs for the Discrete Conley Index Ph. D thesis, Northwesten University, 1998. |
| [15] |
D. S. Richeson,
Connection matrix pairs, Banach Center Publications, 47 (1999), 219-232.
|
| [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. |
| [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. |
| [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. |
| [19] |
S. Smale,
Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 797-817.
doi: 10.1090/S0002-9904-1967-11798-1. |
| [20] |
M. Shub and D. Sullivan,
Homology and dynamical systems, Topology, 14 (1975), 109-132.
doi: 10.1016/0040-9383(75)90022-1. |
| [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. |
show all references
References:
| [1] |
P. Bartiomiejczyk and Z. Dzedzej,
Connection Matrix theory for discrete dynamical systems, Banach Center Publications, 47 (1999), 67-78.
|
| [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.
|
| [3] |
R. Bowen,
Topological entropy and Axiom A, Proc. Sympos. Pure Math., 14 (1970), 23-41.
|
| [4] |
R. Bowen and J. Franks,
Homology for zero-dimensional nonwandering sets, Annals of Mathematics, 106 (1977), 73-92.
doi: 10.2307/1971159. |
| [5] |
C. Conley,
Isolated Invariant Sets and Morse Index CBMS Regional Conference Series in Mathematics, n. 38, American Mathematical Society, Providence, R. I. , 1978. |
| [6] |
J. M. Franks,
Homology and Dynamical Systems, CBMS Regional Conference Series in Mathematics, n. 49, American Mathematical Society, Providence, R. I. , 1982. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [13] |
C. McCord and J. F. Reineck,
Connection matrices and transition matrices, Banach Center Publications, 47 (1999), 41-55.
|
| [14] |
D. S. Richeson,
Connection Matrix Pairs for the Discrete Conley Index Ph. D thesis, Northwesten University, 1998. |
| [15] |
D. S. Richeson,
Connection matrix pairs, Banach Center Publications, 47 (1999), 219-232.
|
| [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. |
| [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. |
| [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. |
| [19] |
S. Smale,
Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 797-817.
doi: 10.1090/S0002-9904-1967-11798-1. |
| [20] |
M. Shub and D. Sullivan,
Homology and dynamical systems, Topology, 14 (1975), 109-132.
doi: 10.1016/0040-9383(75)90022-1. |
| [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. |
Figure 2.
Graded module braids with endomorphisms 1
Figure 3.
Graded module braids with endomorphisms 2
Figure 4.
Smale's Horseshoe
Figure 5.
fitted diffeomorphism
Figure 6.
Smale diffeomorphism
Figure 7.
Connection matrix pair for zero-dimensional basic sets decomposition
Figure 8.
Representation of a connection matrix pair for a zero-dimensional basic set decomposition
Figure 9.
Fitted diffeomorphism
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