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March  2016, 36(3): 1629-1647. doi: 10.3934/dcds.2016.36.1629

On the shape Conley index theory of semiflows on complete metric spaces

Jintao Wang 1, , Desheng Li 2, and  Jinqiao Duan 3,

1. 

Department of Math., School of Science, Tianjin University, Tianjin 300072, China
2. 

Department of Mathematics, School of Science, Tianjin University, Tianjin, 300072
3. 

Department of Applied Math., Illinois Institute of Technology, Chicago IL 60616, United States

Received  December 2014 Revised  June 2015 Published  August 2015

In this work we develop the shape Conley index theory for local semiflows on complete metric spaces by using a weaker notion of shape index pairs. This allows us to calculate the shape index of a compact isolated invariant set $K$ by restricting the system on any closed subset that contains a local unstable manifold of $K$, and hence significantly increases the flexibility of the calculation of shape indices and Morse equations. In particular, it allows to calculate shape indices and Morse equations for an infinite dimensional system by using only the unstable manifolds of the invariant sets, without requiring the system to be two-sided on the unstable manifolds.
Keywords: shape index, shape index pair, local semiflow, Morse equation., Complete metric space.
Mathematics Subject Classification: 37B30, 37D45, 37C7.
Citation: 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
References:
[1]

K. Borsuk, Theory of Shape, Monografie Matematyczne, Tom 59 [Mathematical Monographs, Vol. 59], PWN-Polish Scientific Publishers, Warsaw, 1975.  Google Scholar

[2]

C. Conley, Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, RI, 1978.  Google Scholar

[3]

J. Dydak and J. Segal, Shape Theory: An Introduction, Lecture Notes in Math., 688, Springer-Verlag, Berlin, 1978.  Google Scholar

[4]

A. Giraldo, M. A. Morón, F. R. Ruiz del Portal and J. M. R. Sanjurjo, Shape of global attractors in topological spaces, Nonlinear Anal., 60 (2005), 837-847. doi: 10.1016/j.na.2004.03.036.  Google Scholar

[5]

A. Giraldo, R. Jiménez, M. A. Morón, F. R. Ruiz del Portal and J. M. R. Sanjurjo, Pointed shape and global attractors for metrizable spaces, Topology Appl., 158 (2011), 167-176. doi: 10.1016/j.topol.2010.08.015.  Google Scholar

[6]

A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.  Google Scholar

[7]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[8]

L. Kapitanski and I. Rodnianski, Shape and Morse theory of attractors, Comm. Pure Appl. Math., 53 (2000), 218-242. doi: 10.1002/(SICI)1097-0312(200002)53:2<218::AID-CPA2>3.0.CO;2-W.  Google Scholar

[9]

D. Li, Morse theory of attractors via Lyapunov functions, preprint,, , ().   Google Scholar

[10]

D. Li, G. Shi and X. Song, A linking theory for dynamical systems with applications to PDEs,, , ().   Google Scholar

[11]

S. Mardešić and J. Segal, Shape Theory - The Inverse System Approach, North-Holland Mathematical Library, 26, North-Holland, Amsterdam-New York, 1982.  Google Scholar

[12]

M. Mrozek, Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces, Fund. Math., 145 (1994), 15-37.  Google Scholar

[13]

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

[14]

K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-72833-4.  Google Scholar

[15]

J. J. Sánchez-Gabites, An approach to the Conley shape index without index pairs, Rev. Mat. Complut., 24 (2011), 95-114. doi: 10.1007/s13163-010-0031-x.  Google Scholar

[16]

J. M. R. Sanjurjo, Morse equation and unstable manifolds of isolated invariant set, Nonlinearity, 16 (2003), 1435-1448. doi: 10.1088/0951-7715/16/4/314.  Google Scholar

[17]

J. M. R. Sanjurjo, Shape and Conley index of attractors and isolated invariant sets, Progress Nonlinear Diff. Eqns., 75 (2008), 393-406. doi: 10.1007/978-3-7643-8482-1_29.  Google Scholar

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R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

show all references

References:
[1]

K. Borsuk, Theory of Shape, Monografie Matematyczne, Tom 59 [Mathematical Monographs, Vol. 59], PWN-Polish Scientific Publishers, Warsaw, 1975.  Google Scholar

[2]

C. Conley, Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, RI, 1978.  Google Scholar

[3]

J. Dydak and J. Segal, Shape Theory: An Introduction, Lecture Notes in Math., 688, Springer-Verlag, Berlin, 1978.  Google Scholar

[4]

A. Giraldo, M. A. Morón, F. R. Ruiz del Portal and J. M. R. Sanjurjo, Shape of global attractors in topological spaces, Nonlinear Anal., 60 (2005), 837-847. doi: 10.1016/j.na.2004.03.036.  Google Scholar

[5]

A. Giraldo, R. Jiménez, M. A. Morón, F. R. Ruiz del Portal and J. M. R. Sanjurjo, Pointed shape and global attractors for metrizable spaces, Topology Appl., 158 (2011), 167-176. doi: 10.1016/j.topol.2010.08.015.  Google Scholar

[6]

A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.  Google Scholar

[7]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[8]

L. Kapitanski and I. Rodnianski, Shape and Morse theory of attractors, Comm. Pure Appl. Math., 53 (2000), 218-242. doi: 10.1002/(SICI)1097-0312(200002)53:2<218::AID-CPA2>3.0.CO;2-W.  Google Scholar

[9]

D. Li, Morse theory of attractors via Lyapunov functions, preprint,, , ().   Google Scholar

[10]

D. Li, G. Shi and X. Song, A linking theory for dynamical systems with applications to PDEs,, , ().   Google Scholar

[11]

S. Mardešić and J. Segal, Shape Theory - The Inverse System Approach, North-Holland Mathematical Library, 26, North-Holland, Amsterdam-New York, 1982.  Google Scholar

[12]

M. Mrozek, Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces, Fund. Math., 145 (1994), 15-37.  Google Scholar

[13]

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

[14]

K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-72833-4.  Google Scholar

[15]

J. J. Sánchez-Gabites, An approach to the Conley shape index without index pairs, Rev. Mat. Complut., 24 (2011), 95-114. doi: 10.1007/s13163-010-0031-x.  Google Scholar

[16]

J. M. R. Sanjurjo, Morse equation and unstable manifolds of isolated invariant set, Nonlinearity, 16 (2003), 1435-1448. doi: 10.1088/0951-7715/16/4/314.  Google Scholar

[17]

J. M. R. Sanjurjo, Shape and Conley index of attractors and isolated invariant sets, Progress Nonlinear Diff. Eqns., 75 (2008), 393-406. doi: 10.1007/978-3-7643-8482-1_29.  Google Scholar

[18]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

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