Chaotic behaviour of hyperbolic dynamical systems in a Banach space
In this thesis, C2 maps and semi-flows on separable Banach spaces with invariant ergodic Borel probability measure are considered. By assuming the absence of zero Lyapunov exponents for the discrete case and at most one central direction for the continuous case respectively, there exist periodic orbits and horseshoes. Katok originally established these results for diffeomorphisms on compact manifolds in . In 2011 and 2012, Lian and Young had extended these results for maps and semiflows on infinite dimensional Hilbert space in  and  respectively. In these three papers, they all have inner product structure of tangent space of each point in the domain. In order to overcome the impact of the absence of inner product, two tools were reconstructed under Banach space setting. A measurable Lyapunov chart was reestablished, the invariant manifolds theory was reconstructed to fit the setting of Banach spaces with changing norms along orbits. Using these tools, the existence of periodic orbits and horseshoes for maps and semiflows on Banach spaces was proved.
- Mathematical Sciences
Rights holder© Xiao Ma
NotesA Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.
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