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An Index Theorem in Relative K-Theory for First-Order Systems

Robert Skiba, Daniel Strzelecki, Nils Waterstraat

Abstract

Motivated by bifurcation of branches of homoclinic orbits of dynamical systems, we consider families of first-order equations on the real line and introduce a generalisation of previous index theorems by Pejsachowicz, and by Hu and Portaluri. The main novelties of our approach firstly concern the analytical setting, where we lift the common assumption that the equations are asymptotically hyperbolic. Secondly, we consider general compact parameter spaces instead of a single parameter, which results in a remarkably simple index formula in relative $K$-theory.

An Index Theorem in Relative K-Theory for First-Order Systems

Abstract

Motivated by bifurcation of branches of homoclinic orbits of dynamical systems, we consider families of first-order equations on the real line and introduce a generalisation of previous index theorems by Pejsachowicz, and by Hu and Portaluri. The main novelties of our approach firstly concern the analytical setting, where we lift the common assumption that the equations are asymptotically hyperbolic. Secondly, we consider general compact parameter spaces instead of a single parameter, which results in a remarkably simple index formula in relative -theory.
Paper Structure (18 sections, 23 theorems, 216 equations)

This paper contains 18 sections, 23 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. First Order Equations and Exponential Dichotomies
  4. The Index Bundle in Relative $K$-Theory
  5. Relative $K$-Theory
  6. The Index Bundle
  7. The Family Index Theorem
  8. The Differential Operators and their Adjoints
  9. Index Theorem and Corollaries
  10. Comparison to the Work of Pejsachowicz and Hu-Portaluri
  11. Pejsachowicz' Index Formula
  12. Hu and Portaluri's Index Formula
  13. Proof of the Index Theorem
  14. An Example
  15. Bifurcation of Homoclinic Orbits
  16. ...and 3 more sections

Key Result

Theorem 1

Let $X,Y$ be Banach spaces and $G:[a,b]\times X\rightarrow Y$ a $C^1$ map such that $G(\lambda,0)=0$ for all $\lambda\in[a,b]$. Assume that the derivatives $L_\lambda:=D_uG(\lambda,0):X\rightarrow Y$ are Fredholm of index $0$ for all $\lambda\in[a,b]$, as well as invertible for $\lambda\in \{a,b\}$. i.e., there is some $\lambda^\ast\in(a,b)$ such that in every neighbourhood of $(\lambda^\ast,0)$ i

Theorems & Definitions (34)

  • Theorem
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Theorem 3.1
  • Lemma 3.2
  • ...and 24 more