236 research outputs found
On critical renormalization of complex polynomials
Holomorphic renormalization plays an important role in complex polynomial
dynamics. We consider certain conditions guaranteeing that a polynomial which
does not admit a polynomial-like connected Julia set still admits an invariant
continuum on which it is topologically conjugate to a lower degree polynomial.
This invariant continuum may contain extra critical points of the original
polynomial that are not visible in the dynamical plane of the conjugate
polynomial. Thus, we extend the notions of holomorphic renormalization and
polynomial-like maps and describe a setup where new generalized versions of
these notions are applicable and yield useful topological conjugacies.Comment: 25 pages, 3 figure
A finite subdivision rule for the n-dimensional torus
Cannon, Floyd, and Parry have studied subdivisions of the 2-sphere
extensively, especially those corresponding to 3-manifolds, in an attempt to
prove Cannon's conjecture. There has been a recent interest in generalizing
some of their tools, such as extremal length, to higher dimensions. We define
finite subdivision rules of dimension n, and find an n-1-dimensional finite
subdivision rule for the n-dimensional torus, using a well-known simplicial
decomposition of the hypercube. We hope to expand on this and find finite
subdivision rules for many higher-dimensional manifolds, including hyperbolic
n-manifolds.Comment: Accepted by Geometriae Dedicata; ublished version available onlin
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