From rubber bands to rational maps: a research report

被引:10
作者
Thurston, Dylan P. [1 ]
机构
[1] Indiana Univ, Dept Math, 831 E Third St, Bloomington, IN 47405 USA
基金
美国国家科学基金会;
关键词
Complex dynamics; Dirichlet energy; Elastic graphs; Extremal length; Measured foliations; Riemann surfaces; Rational maps; Surface embeddings; QUADRATIC-DIFFERENTIALS; MEASURED FOLIATIONS; TRAIN TRACKS; GEOMETRY; MATINGS; THEOREM; PROOF;
D O I
10.1186/s40687-015-0039-4
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
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. On 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.
引用
收藏
页数:49
相关论文
共 42 条
[1]  
Agol I, 2011, CONTEMP MATH, V560, P1
[2]  
[Anonymous], ARXIV10093647
[3]  
[Anonymous], 2001, CAMBRIDGE TRACTS MAT
[4]  
[Anonymous], 1970, CONVEX ANAL
[5]  
[Anonymous], 2000, London Math. Soc. Lecture Note Ser.
[6]  
[Anonymous], 2015, ARXIV150705294
[7]   TRAIN TRACKS AND AUTOMORPHISMS OF FREE GROUPS [J].
BESTVINA, M ;
HANDEL, M .
ANNALS OF MATHEMATICS, 1992, 135 (01) :1-51
[8]   A Bers-like proof of the existence of train tracks for free group automorphisms [J].
Bestvina, Mladen .
FUNDAMENTA MATHEMATICAE, 2011, 214 (01) :1-12
[9]  
Bonnot S, 2012, MOSC MATH J, V12, P747
[10]  
Brooks R.L., 1940, Duke Math. J., V7, P312, DOI 10.1215/S0012-7094-40-00718-9