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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

 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.
Citation: Jintao Wang, Desheng Li, Jinqiao Duan. On the shape Conley index theory of semiflows on complete metric spaces. Discrete and 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. [2] C. Conley, Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, RI, 1978. [3] J. Dydak and J. Segal, Shape Theory: An Introduction, Lecture Notes in Math., 688, Springer-Verlag, Berlin, 1978. [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. [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. [6] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. [7] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [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. [9] D. Li, Morse theory of attractors via Lyapunov functions, preprint,, , (). [10] D. Li, G. Shi and X. Song, A linking theory for dynamical systems with applications to PDEs,, , (). [11] S. Mardešić and J. Segal, Shape Theory - The Inverse System Approach, North-Holland Mathematical Library, 26, North-Holland, Amsterdam-New York, 1982. [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. [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. [14] K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-72833-4. [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. [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. [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. [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.

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##### References:
 [1] K. Borsuk, Theory of Shape, Monografie Matematyczne, Tom 59 [Mathematical Monographs, Vol. 59], PWN-Polish Scientific Publishers, Warsaw, 1975. [2] C. Conley, Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, RI, 1978. [3] J. Dydak and J. Segal, Shape Theory: An Introduction, Lecture Notes in Math., 688, Springer-Verlag, Berlin, 1978. [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. [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. [6] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. [7] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [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. [9] D. Li, Morse theory of attractors via Lyapunov functions, preprint,, , (). [10] D. Li, G. Shi and X. Song, A linking theory for dynamical systems with applications to PDEs,, , (). [11] S. Mardešić and J. Segal, Shape Theory - The Inverse System Approach, North-Holland Mathematical Library, 26, North-Holland, Amsterdam-New York, 1982. [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. [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. [14] K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-72833-4. [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. [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. [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. [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.
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