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Title: Chaotic behaviour of hyperbolic dynamical systems in a Banach space
Author: Ma, Xiao
ISNI:       0000 0004 7970 961X
Awarding Body: Loughborough University
Current Institution: Loughborough University
Date of Award: 2017
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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 [12]. In 2011 and 2012, Lian and Young had extended these results for maps and semiflows on infinite dimensional Hilbert space in [17] and [18] 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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematical Sciences not elsewhere classified ; Lyapunov exponents ; Periodic orbits ; Horseshoes ; Banach space