2,262 research outputs found
A Combinatorial Analog of a Theorem of F.J.Dyson
Tucker's Lemma is a combinatorial analog of the Borsuk-Ulam theorem and the
case n=2 was proposed by Tucker in 1945. Numerous generalizations and
applications of the Lemma have appeared since then. In 2006 Meunier proved the
Lemma in its full generality in his Ph.D. thesis. There are generalizations and
extensions of the Borsuk-Ulam theorem that do not yet have combinatorial
analogs. In this note, we give a combinatorial analog of a result of Freeman J.
Dyson and show that our result is equivalent to Dyson's theorem. As with
Tucker's Lemma, we hope that this will lead to generalizations and applications
and ultimately a combinatorial analog of Yang's theorem of which both
Borsuk-Ulam and Dyson are special cases.Comment: Original version: 7 pages, 2 figures. Revised version: 12 pages, 4
figures, revised proofs. Final revised version: 9 pages, 2 figures, revised
proof
Bifurcations in the Space of Exponential Maps
This article investigates the parameter space of the exponential family
. We prove that the boundary (in \C) of every
hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as
well as Baker and Rippon. In fact, we prove the stronger statement that the
exponential bifurcation locus is connected in \C, which is an analog of
Douady and Hubbard's celebrated theorem that the Mandelbrot set is connected.
We show furthermore that is not accessible through any nonhyperbolic
("queer") stable component.
The main part of the argument consists of demonstrating a general "Squeezing
Lemma", which controls the structure of parameter space near infinity. We also
prove a second conjecture of Eremenko and Lyubich concerning bifurcation trees
of hyperbolic components.Comment: 29 pages, 3 figures. The main change in the new version is the
introduction of Theorem 1.1 on the connectivity of the bifurcation locus,
which follows from the results of the original version but was not explicitly
stated. Also, some small revisions have been made and references update
Latt\`es maps and combinatorial expansion
A Latt\`es map is a
rational map that is obtained from a finite quotient of a conformal torus
endomorphism. We characterize Latt\`es maps by their combinatorial expansion
behavior.Comment: 41 pages, 3 figures. arXiv admin note: text overlap with
arXiv:1109.2980; and with arXiv:1009.3647 by other author
Quasisymmetric parametrizations of two-dimensional metric spheres
We study metric spaces homeomorphic to the 2-sphere, and find conditions
under which they are quasisymmetrically homeomorphic to the standard 2-sphere.
As an application of our main theorem we show that an Ahlfors 2-regular,
linearly locally contractible metric 2-sphere is quasisymmetrically
homeomorphic to the standard 2-sphere, answering a question of Heinonen and
Semmes
Thurston obstructions and Ahlfors regular conformal dimension
Let be an expanding branched covering map of the sphere to
itself with finite postcritical set . Associated to is a canonical
quasisymmetry class \GGG(f) of Ahlfors regular metrics on the sphere in which
the dynamics is (non-classically) conformal. We show \inf_{X \in \GGG(f)}
\hdim(X) \geq Q(f)=\inf_\Gamma \{Q \geq 2: \lambda(f_{\Gamma,Q}) \geq 1\}.
The infimum is over all multicurves . The map
is defined by where the second sum is over all preimages
of freely homotopic to in , and is its Perron-Frobenius leading eigenvalue. This
generalizes Thurston's observation that if , then there is no
-invariant classical conformal structure.Comment: Minor revisions are mad
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