70 research outputs found
Algorithmic aspects of branched coverings
This is the announcement, and the long summary, of a series of articles on
the algorithmic study of Thurston maps. We describe branched coverings of the
sphere in terms of group-theoretical objects called bisets, and develop a
theory of decompositions of bisets.
We introduce a canonical "Levy" decomposition of an arbitrary Thurston map
into homeomorphisms, metrically-expanding maps and maps doubly covered by torus
endomorphisms. The homeomorphisms decompose themselves into finite-order and
pseudo-Anosov maps, and the expanding maps decompose themselves into rational
maps.
As an outcome, we prove that it is decidable when two Thurston maps are
equivalent. We also show that the decompositions above are computable, both in
theory and in practice.Comment: 60-page announcement of 5-part text, to apper in Ann. Fac. Sci.
Toulouse. Minor typos corrected, and major rewrite of section 7.8, which was
studying a different map than claime
From rubber bands to rational maps: A research report
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
Injections of mapping class groups
We construct new monomorphisms between mapping class groups of surfaces. The
first family of examples injects the mapping class group of a closed surface
into that of a different closed surface. The second family of examples are
defined on mapping class groups of once-punctured surfaces and have quite
curious behaviour. For instance, some pseudo-Anosov elements are mapped to
multi-twists. Neither of these two types of phenomena were previously known to
be possible although the constructions are elementary
The Burau estimate for the entropy of a braid
The topological entropy of a braid is the infimum of the entropies of all
homeomorphisms of the disc which have a finite invariant set represented by the
braid. When the isotopy class represented by the braid is pseudo-Anosov or is
reducible with a pseudo-Anosov component, this entropy is positive. Fried and
Kolev proved that the entropy is bounded below by the logarithm of the spectral
radius of the braid's Burau matrix, , after substituting a complex number
of modulus~1 in place of . In this paper we show that for a pseudo-Anosov
braid the estimate is sharp for the substitution of a root of unity if and only
if it is sharp for . Further, this happens if and only if the invariant
foliations of the pseudo-Anosov map have odd order singularities at the strings
of the braid and all interior singularities have even order. An analogous
theorem for reducible braids is also proved.Comment: 28 pages, 8 figure
Thurston's theorem and the Nielsen-Thurston classification via Teichm\"uller's theorem
We give a unified and self-contained proof of the Nielsen-Thurston
classification theorem from the theory of mapping class groups and Thurston's
characterization of rational maps from the theory of complex dynamics (plus
various extensions of these). Our proof follows Bers' proof of the
Nielsen-Thurston classification.Comment: 36 pages, 0 figure
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Billiards and Teichmüller Curves on Hilbert Modular Surfaces
This paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two. These Teichmüller curves lie on Hilbert modular surfaces parameterizing Abelian varieties with real multiplication. Explicit examples, constructed from L-shaped polygons, give billiard tables with optimal dynamical properties.Mathematic
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