Iterated monodromy groups of Chebyshev-like maps on $\mathbb{C}^n$
Every affine Weyl group appears as the iterated monodromy group of a Chebyshev-like polynomial self-map of $\mathbb{C}^n$.
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arXiv subjects
Joshua P. Bowman
Publications and source records attributed to Joshua P. Bowman.
Geometry and Algebra of the Deltoid Map
The geometry of the deltoid curve gives rise to a self-map of $\mathbb{C}^2$ that is expressed in coordinates by $f(x,y) = (y^2 - 2x, x^2 - 2y)$. This is one in a family of maps that generalize Chebyshev polynomials to several variables. We use this example to illustrate two important objects in complex dynamics: the Julia set and the iterated monodromy group.
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Finiteness conditions on translation surfaces
We consider various metric and analytic notions of finiteness on translation surfaces. The Veech group of a surface is discrete if the surface has finite area or is totally bounded.
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Wild singularities of flat surfaces
We consider flat surfaces and the points of their metric completions, particularly the singularities to which the flat structure of the surface does not extend. The local behavior near a singular point x can be partially described by a topological space L(x) which captures all the ways that x can be "approached linearly". The homeomorphism type of L(x) is an affine invariant. When x is not a cone point or an infinite-angle singularity, we say it is wild; in this case it is necessary to add further metric data to L(x) to get a quantitative description of the surface near x.
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The complete family of Arnoux-Yoccoz surfaces
The family of translation surfaces $(X_g,ω_g)$ constructed by Arnoux and Yoccoz from self-similar interval exchange maps encompasses one example from each genus $g$ greater than or equal to $3$. We triangulate these surfaces and deduce general properties they share. The surfaces $(X_g,ω_g)$ converge to a surface $(X_\infty,ω_\infty)$ of infinite genus and finite area. We study the exchange on infinitely many intervals that arises from the vertical flow on $(X_\infty,ω_\infty)$ and compute the affine group of $(X_\infty,ω_\infty)$, which has an index $2$ cyclic subgroup generated by a hyperbolic element.
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Orientation-reversing involutions of the genus 3 Arnoux-Yoccoz surface and related surfaces
We present a new description of the genus 3 Arnoux--Yoccoz translation surface in terms of its Delaunay polygons and show that, up to affine equivalence, it belongs to two families of surfaces whose isometry groups include the dihedral group of the square.
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