Browsing by Author "Roeder, Roland K."

Now showing 1 - 4 of 4
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    Complex perspective for the projective heat map acting on pentagons
    (AMS, 2017) Kaschner, Scott R.; Roeder, Roland K.; Mathematical Sciences, School of Science
    We place Schwartz's work on the real dynamics of the projective heat map H into the complex perspective by computing its rst dynamical degree and gleaning some corollaries about the dynamics of H.
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    COMPUTING DYNAMICAL DEGREES OF RATIONAL MAPS ON MODULI SPACE
    (Cambridge, 2015) Koch, Sarah; Roeder, Roland K.; Department of Mathematical Sciences, School of Science
    The dynamical degrees of a rational map f:X⇢X are fundamental invariants describing the rate of growth of the action of iterates of f on the cohomology of X. When f has non-empty indeterminacy set, these quantities can be very difficult to determine. We study rational maps f:XN⇢XN, where XN is isomorphic to the Deligne–Mumford compactification M¯¯¯¯0,N+3. We exploit the stratified structure of XN to provide new examples of rational maps, in arbitrary dimension, for which the action on cohomology behaves functorially under iteration. From this, all dynamical degrees can be readily computed (given enough book-keeping and computing time). In this paper, we explicitly compute all of the dynamical degrees for all such maps f:XN⇢XN, where dim(XN)≤3 and the first dynamical degrees for the mappings where dim(XN)≤5. These examples naturally arise in the setting of Thurston’s topological characterization of rational maps.
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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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    Pulling back singularities of codimension one objects
    (AMS, 2019) Giraldo, Luis; Roeder, Roland K.; Mathematical Sciences, School of Science
    We prove that the preimage of a germ of a singular analytic hypersurface under a germ of a finite holomorphic map $ g: (\mathbb{C}^n,{\bm 0}) \rightarrow (\mathbb{C}^n,{\bm 0})$ is again singular. This provides a generalization of previous results of this nature by Ebenfelt-Rothschild [Comm. Anal.Geom. 15 (2007), no. 2, 491-507], Lebl [ArXiv preprint https://arxiv.orglabs/0812-2498], and Denkowski [Manuscripta Math. 149 (2016), no. 1-2, 83-91]. The same statement is proved for pullbacks of singular codimension one holomorphic foliations.