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Critical dynamics of variable‐separated affine correspondences
Author(s) -
Ingram Patrick
Publication year - 2017
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12045
Subject(s) - mathematics , bounded function , affine transformation , holomorphic function , property (philosophy) , pure mathematics , orbit (dynamics) , space (punctuation) , variable (mathematics) , finite set , affine plane (incidence geometry) , combinatorics , mathematical analysis , computer science , philosophy , plane curve , epistemology , engineering , aerospace engineering , operating system
We examine affine correspondences of the form g ( y ) = f ( x ) , for f and g polynomials satisfying deg ( g ) < deg ( f ) , with the property that every critical point of the correspondence admits at least one finite forward orbit. In the case g ( y ) = y , this reduces to the study of post‐critically finite polynomials, and our main result extends earlier finiteness results of the author. Specifically, we show that the collection of such correspondences of a given bidegree coincides with a subset of the parameter space of bounded Weil height. We also show that there are no non‐trivial holomorphic families of correspondences with the above‐described property.