# American Institute of Mathematical Sciences

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March  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

* 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.

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.
Braid diagram
Graded module braids with endomorphisms 1
Graded module braids with endomorphisms 2
Smale's Horseshoe
fitted diffeomorphism
Smale diffeomorphism
Connection matrix pair for zero-dimensional basic sets decomposition
Representation of a connection matrix pair for a zero-dimensional basic set decomposition
Fitted diffeomorphism
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