117,171 research outputs found
Combinatorial rigidity of multicritical maps
We combine the KSS nest constructed by Kozlovski, Shen and van Strien, and
the analytic method proposed by Avila, Kahn, Lyubich and Shen to prove the
combinatorial rigidity of multicritical maps.Comment: 30 pages, 8 figure
Census of Planar Maps: From the One-Matrix Model Solution to a Combinatorial Proof
We consider the problem of enumeration of planar maps and revisit its
one-matrix model solution in the light of recent combinatorial techniques
involving conjugated trees. We adapt and generalize these techniques so as to
give an alternative and purely combinatorial solution to the problem of
counting arbitrary planar maps with prescribed vertex degrees.Comment: 29 pages, 14 figures, tex, harvmac, eps
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
On the Hyperbolicity of Lorenz Renormalization
We consider infinitely renormalizable Lorenz maps with real critical exponent
and combinatorial type which is monotone and satisfies a long return
condition. For these combinatorial types we prove the existence of periodic
points of the renormalization operator, and that each map in the limit set of
renormalization has an associated unstable manifold. An unstable manifold
defines a family of Lorenz maps and we prove that each infinitely
renormalizable combinatorial type (satisfying the above conditions) has a
unique representative within such a family. We also prove that each infinitely
renormalizable map has no wandering intervals and that the closure of the
forward orbits of its critical values is a Cantor attractor of measure zero.Comment: 63 pages; 10 figure
Renormalization theory for multimodal maps
We study the dynamics of the renormalization operator for multimodal maps. In
particular, we prove the exponential convergence of this operator for
infinitely renormalizable maps with same bounded combinatorial type.Comment: 37 pages, 4 figure
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