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Dynamical Classification of a Family of Birational Maps of \({\mathbb {C}}^2\) via Algebraic Entropy

  • Sundus Zafar
  • Anna Cima
Article
  • 2 Downloads

Abstract

This work dynamically classifies a 9-parametric family of birational maps \(f: {\mathbb {C}}^2 \rightarrow {\mathbb {C}}^2\). From the sequence of the degrees \(d_n\) of the iterates of f,  we find the dynamical degree \(\delta (f)\) of f. We identify when \(d_n\) grows periodically, linearly, quadratically or exponentially. The considered family includes the birational maps studied by Bedford and Kim (Mich Math J 54:647–670, 2006) as one of its subfamilies.

Keywords

Birational maps Algebraic entropy First integrals Fibrations Blowing-up Integrability Periodicity Chaos 

Mathematics Subject Classification

14E05 26C15 34K19 28D20 37C15 39A23 39A45 

Notes

Acknowledgements

The first author is supported by Ministry of Economy, Industry and Competitiveness of the Spanish Government through Grants MINECO/FEDER MTM2016-77278 and also supported by the grant 2017-SGR-1617 from AGAUR, Generalitat de Catalunya.

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Authors and Affiliations

  1. 1.Universitat Autonoma de BarcelonaBarcelonaSpain

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