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Session 16. Ergodic Theory and Dynamical Systems
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On the dichotomy spectrum |
Christian Pötzsche, Alpen-Adria Universität Klagenfurt, Austria
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When dealing with nonautonomous dynamical systems, it is well-known
that eigenvalues yield no information on the stability or
hyperbolicity of linear systems. Therefore, several more appropriate
spectral notions were developped.
Our particular focus is on the dichotomy spectrum (also
called dynamical or Sacker-Sell spectrum). It is a
crucial notion in the theory of dynamical systems, since it contains
information on stability, as well as appropriate robustness
properties. However, recent applications in nonautonomous
bifurcation theory showed that a detailed insight into the fine
structure of this spectral notion is necessary. On this basis, we
explore a helpful connection between the dichotomy spectrum and
operator theory. It relates the asymptotic behavior of linear
nonautonomous equations to the (approximate) point, surjectivity and
Fredholm spectra of weighted shifts. This link yields several
dynamically meaningful subsets of the dichotomy spectrum, which not
only allows to classify and detect bifurcations, but also simplifies
proofs for results on the long term behavior. Moreover, robustness
properties of the dichotomy spectrum
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References- B. Aulbach and N. Van Minh, The
concept of spectral dichotomy for linear difference equations
II , J. Difference Equ. Appl. 2 (1996), 251-262.
- C. Pötzsche, Fine structure of the
dichotomy spectrum , Integral Equations and Operator Theory
73 (2012), no. 1, 107-151.
- R.J. Sacker and G.R. Sell, A spectral
theory for linear differential systems , J. Differ. Equations
27 (1978), 320-358.
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