7,475 research outputs found

    From rubber bands to rational maps: A research report

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    This research report outlines work, partially joint with Jeremy Kahn and Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal surfaces with boundary. One one hand, this lets us tell when one rubber band network is looser than another, and on the other hand tell when one conformal surface embeds in another. We apply this to give a new characterization of hyperbolic critically finite rational maps among branched self-coverings of the sphere, by a positive criterion: a branched covering is equivalent to a hyperbolic rational map if and only if there is an elastic graph with a particular "self-embedding" property. This complements the earlier negative criterion of W. Thurston.Comment: 52 pages, numerous figures. v2: New example

    A Combinatorial classification of postcritically fixed Newton maps

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    We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps. A fundamental ingredient is the proof that for every Newton map (postcritically finite or not) every connected component of the basin of an attracting fixed point can be connected to ∞\infty through a finite chain of such components.Comment: 37 pages, 5 figures, published in Ergodic Theory and Dynamical Systems (2018). This is the final author file before publication. Text overlap with earlier arxiv file observed by arxiv system relates to an earlier version that was erroneously uploaded separately. arXiv admin note: text overlap with arXiv:math/070117

    Domino tilings and related models: space of configurations of domains with holes

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    We first prove that the set of domino tilings of a fixed finite figure is a distributive lattice, even in the case when the figure has holes. We then give a geometrical interpretation of the order given by this lattice, using (not necessarily local) transformations called {\em flips}. This study allows us to formulate an exhaustive generation algorithm and a uniform random sampling algorithm. We finally extend these results to other types of tilings (calisson tilings, tilings with bicolored Wang tiles).Comment: 17 pages, 11 figure
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