Melbourne, Ian

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Melbourne, Ian
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Melbourne, Ian
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    Item type:Publication,
    Statistics of rare events in infinite ergodic theory
    (2016-01-20)
    Kautzsch, Johannes Karl
    ;
    Melbourne, Ian
    ;
    Kesseböhmer, Marc
    ;
    Kesseböhmer, Marc
    This thesis studies the asymptotic behaviour of the transfer operator for systems having a sigma-finite invariant measure. Firstly, a family of Markov interval maps interpolating between the Tent map and the Farey map is considered. This part considers distributional convergence of observables which are integrable, but posses singularities. It turns out that the limiting behaviour on the omega-limit set of the pole depends on the diophantine properties of the pole. Yet, distributional convergence can still be obtained on compact subsets, that do not intersect the omega-limit set of the pole and that are bounded away from the indifferent fixed point. Secondly, this thesis focuses on how to modify the transformation in general and its wandering rate in particular. It turns out that additional assumptions may be required, if the wandering rate is no longer slowly varying but regularly varying. In this part of the thesis a family of non-uniformly hyperbolic maps, known as the alpha-Farey maps is considered.
    doctoral thesis
      542  264
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    Item type:Publication,
    Ergodic theory of nonlinear waves in discrete and continuous excitable media
    (2021-08-25)
    Ulbrich, Dennis
    ;
    Melbourne, Ian
    ;
    Rademacher, Jens
    ;
    Rademacher, Jens
    In this thesis, we analyze discrete and continuous models of excitable media with the intention to reveal similarities between both approaches in terms of wave propagation and interaction. While the discrete perspective is represented by the one-dimensional Greenberg-Hastings cellular automata (GHCA), as a continuous model we consider the $\theta$-equations which are basic partial differential equations (PDE) for pure phase dynamics. On the one hand, qualitatively, collision and annihilation of waves can be observed in both models in striking resemblance. However, on the other hand, it turns out that a quantitative comparison of discrete and continuous wave interactions is limited due to weak wave interactions in the PDE. Specifically, complexity considerations show that a direct comparison of discrete and continuous strong wave interactions is problematic.
    doctoral thesis
      540  687