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On the multipliers at fixed points of quadratic self-maps of the projective plane with an invariant line. (English) Zbl 1434.37027
Summary: This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely determined, up to linear equivalence, by the collection of these eigenvalues. Under the natural correspondence between quadratic rational maps of \({\mathbb{P}}^2\) and quadratic homogeneous vector fields on \({\mathbb{C}}^3\), the algebraic relations among multipliers translate to algebraic relations among the Kowalevski exponents of a vector field. As an application of our results, we describe the sets of integers that appear as the Kowalevski exponents of a class of quadratic homogeneous vector fields on \({\mathbb{C}}^3\) having exclusively single-valued solutions.

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37J70 Completely integrable discrete dynamical systems
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
Full Text: DOI
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