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SPACE OF INITIAL VALUES OF A MAP WITH A QUARTIC INVARIANT

Part of: Complex dynamical systems Curves Difference equations Surfaces and higher-dimensional varieties Difference and functional equations Birational geometry

Published online by Cambridge University Press:  05 October 2020

GIORGIO GUBBIOTTI Open the ORCID record for GIORGIO GUBBIOTTI [Opens in a new window]  and
NALINI JOSHI Open the ORCID record for NALINI JOSHI [Opens in a new window]
GIORGIO GUBBIOTTI*
Affiliation:
School of Mathematics and Statistics, The University of Sydney, NSW2006, Australia
NALINI JOSHI
Affiliation:
School of Mathematics and Statistics, The University of Sydney, NSW2006, Australia e-mail: nalini.joshi@sydney.edu.au
*
e-mail: giorgio.gubbiotti@sydney.edu.au
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Abstract

We compactify and regularise the space of initial values of a planar map with a quartic invariant and use this construction to prove its integrability in the sense of algebraic entropy. The system has certain unusual properties, including a sequence of points of indeterminacy in $\mathbb {P}^{1}\!\times \mathbb {P}^{1}$ . These indeterminacy points lie on a singular fibre of the mapping to a corresponding QRT system and provide the existence of a one-parameter family of special solutions.

Keywords

space of initial valuesintegrabilityKahan discretisationresolution of indeterminacies

MSC classification

Primary: 14E05: Rational and birational maps
Secondary: 14E15: Global theory and resolution of singularities 14H70: Relationships with integrable systems 14J17: Singularities 37F10: Polynomials; rational maps; entire and meromorphic functions 39A10: Difference equations, additive
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 3 , June 2021 , pp. 438 - 449
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The research reported in this paper was supported by the Australian Laureate Fellowship #FL120100094 and Discovery Project #DP190101838 from the Australian Research Council.

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