Discrete conley index theory for zero dimensional basic sets

Ketty A. De Rezende; Mariana G. Villapouca

doi:10.3934/dcds.2017056

Article Contents

2017, Volume 37, Issue 3: 1359-1387. Doi: 10.3934/dcds.2017056

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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
* Corresponding author

Received: April 30, 2016

Revised: August 31, 2016

Published: May 2017

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

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Abstract

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.

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Figure 1. Braid diagram

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Figure 2. Graded module braids with endomorphisms 1

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Figure 3. Graded module braids with endomorphisms 2

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Figure 4. Smale's Horseshoe

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Figure 5. fitted diffeomorphism

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Figure 6. Smale diffeomorphism

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

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

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Figure 9. Fitted diffeomorphism

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