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

Jintao  Wang; Desheng Li; Jinqiao  Duan

doi:10.3934/dcds.2016.36.1629

Article Contents

2016, Volume 36, Issue 3: 1629-1647. Doi: 10.3934/dcds.2016.36.1629

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

1.
Department of Math., School of Science, Tianjin University, Tianjin 300072
2.
Department of Mathematics, School of Science, Tianjin University, Tianjin, 300072
3.
Department of Applied Math., Illinois Institute of Technology, Chicago IL 60616

Received: December 2014

Revised: June 2015

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: 37B30, 37D45, 37C75.

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