Browsing by Author "Geller, William"

Now showing 1 - 4 of 4
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    Coarse entropy of metric spaces
    (Springer, 2024) Geller, William; Misiurewicz, Michał; Sawicki, Damian; Mathematical Sciences, School of Science
    Coarse geometry studies metric spaces on the large scale. The recently introduced notion of coarse entropy is a tool to study dynamics from the coarse point of view. We prove that all isometries of a given metric space have the same coarse entropy and that this value is a coarse invariant. We call this value the coarse entropy of the space and investigate its connections with other properties of the space. We prove that it can only be either zero or infinity, and although for many spaces this dichotomy coincides with the subexponential-exponential growth dichotomy, there is no relation between coarse entropy and volume growth more generally. We completely characterise this dichotomy for spaces with bounded geometry and for quasi-geodesic spaces. As an application, we provide an example where coarse entropy yields an obstruction for a coarse embedding, where such an embedding is not precluded by considerations of volume growth.
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    The Dynamics of Semigroups of Contraction Similarities on the Plane
    (2019-08) Silvestri, Stefano; Perez, Rodrigo; Geller, William; Misiurewicz, Michal; Roeder, Roland K.
    Given a parametrized family of Iterated Function System (IFS) we give sufficient conditions for a parameter on the boundary of the connectedness locus, M, to be accessible from the complement of M. Moreover, we provide a few examples of such parameters and describe how they are connected to Misiurewicz parameter in the Mandelbrot set, i.e. the connectedness locus of the quadratic family z^2+c.
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    The Dynamics of Twisted Tent Maps
    (2013-07-12) Chamblee, Stephen Joseph; Misiurewicz, Michał, 1948-; Roeder, Roland; Geller, William; Eremenko, Alexandre; Mukhin, Evgeny
    This paper is a study of the dynamics of a new family of maps from the complex plane to itself, which we call twisted tent maps. A twisted tent map is a complex generalization of a real tent map. The action of this map can be visualized as the complex scaling of the plane followed by folding the plane once. Most of the time, scaling by a complex number will \twist" the plane, hence the name. The "folding" both breaks analyticity (and even smoothness) and leads to interesting dynamics ranging from easily understood and highly geometric behavior to chaotic behavior and fractals.
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    Farey–Lorenz Permutations for Interval Maps
    (World Scientific, 2018-02) Geller, William; Misiurewicz, Michał; Mathematical Sciences, School of Science
    Lorenz-like maps arise in models of neuron activity, among other places. Motivated by questions about the pattern of neuron firing in such a model, we study periodic orbits and their itineraries for Lorenz-like maps with nondegenerate rotation intervals. We characterize such orbits for the simplest such case and gain substantial information about the general case.