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Zeta Functions and Asymptotic Formulae for Preperiodic Orbits of Hyperbolic Rational Maps
Author(s) -
Waddington Simon
Publication year - 1997
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.3211860116
Subject(s) - mathematics , riemann sphere , julia set , diophantine equation , number theory , degree (music) , pure mathematics , function (biology) , riemann hypothesis , mathematical analysis , riemann surface , physics , acoustics , evolutionary biology , biology
For a hyperbolic rational map R of the Riemann sphere of degree d ≥ 2, restricted to its Julia set J(R ), we define a zeta function ζ R ( s ), which counts the prepenodic orbib of R , according to the weight function |R'| : J(R) → C. An analysis of the analytic domain of ζ R ( s ), using techniques from symbolic dynamics, yields weighted asymptotic formulae for the preperiodic orbits of R . We describe an application to diophantine number theory.