Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity

By Laura Demarco. 2003

122 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0eedb21593b1f6eedb45fd7904ae383a37576f01

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Citations in the corpus

SPECIAL CURVES AND POSTCRITICALLY FINITE POLYNOMIALS (2012, background): “2;t 2) = s 1t 2 s 2t 1. Remark 3.6. For varchimedean, the fact that g v (x;y) is normalized implies (and in fact is equivalent to) the statement that e v is the homogeneous capacity (in the sense of [De2]) of the set K= f(s;t) 2C2 : H0g:This is proved in a slightly more roundabout way in [De2, x4]. ”
Bifurcation currents in holomorphic dynamics on (2007, methodology): “More recently, DeMarco ([8], [9]) has obtained a generalization of this formula to rational maps of P1.”
Dynamics in Several Complex Variables: Endomorphisms of Projective Spaces and Polynomial-like Mappings (2008, result): “T.-C. Dinh UPMC Univ Paris 06, UMR 7586, Institut de Mathématiques de Jussieu, F-75005 Paris, France e-mail: dinh@math.jussieu.fr N. Sibony Université Paris-Sud 11, Mathématique - Bâtiment 425, 91405 Orsay, France e-mail: nessim.sibony@math.u-psud.fr G. Gentili et al. (eds.)”
A finiteness theorem for canonical heights attached to rational maps over function fields (2006, background): “t to −log|F(z)∧w| = (d− 1)logR − Xd i=1 log|z ∧ wi| , which itself is equivalent to (A.12) |F(z) ∧w| = |Res(F)|1/d Yd i=1 |z ∧wi| . 24 MATTHEWBAKER Weverify (A.12)byanexplicit calculation (comparewith[11], Lemma 6.5). Write F = (F 1,F 2) = ( Qd i=1z ∧ ai, Qd j=1z ∧ bj). Since F(bj) = ( Qd i=1ai ∧ bj”
Bifurcation currents in holomorphic dynamics on ${\bf P}^k$ (2005, background): “Remark 4.11 In the one-dimensional case Propositions 4.9 and 4.10 give a new proof of DeMarco’s formula (see [9] Corollary 1.6) since ∫ P1 gF (µf + ω) = 1 2 (log |Res(F )| − 1).”
Distribution of rational maps with a preperiodic critical point (2006, background): “a (ck) = 0}. By summing these currents, we get a current whose support is the bifurcation locus. This is the bifurcation current, originally considered in [DeM1]. It was proved to equal ddcLyap(P) in [DeM2] for families of rational maps. For convenience, we restate and reprove her results in our con”
Bifurcation Currents in Holomorphic Families of Rational Maps (2012, background): “The following result has been originally proved by DeMarco in [20], [21].”