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Dynamical degrees of affine-triangular automorphisms of affine spaces

Part of: Affine geometry Complex dynamical systems

Published online by Cambridge University Press:  01 October 2021

JÉRÉMY BLANC  and
IMMANUEL VAN SANTEN
JÉRÉMY BLANC*
Affiliation:
Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051Basel, Switzerland (e-mail: immanuel.van.santen@math.ch)
IMMANUEL VAN SANTEN
Affiliation:
Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051Basel, Switzerland (e-mail: immanuel.van.santen@math.ch)
*
e-mail: jeremy.blanc@unibas.ch
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Abstract

We study the possible dynamical degrees of automorphisms of the affine space $\mathbb {A}^n$ . In dimension $n=3$ , we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalizes the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space $\mathbb {A}^n$ for some n, and we give the best possible n for quadratic integers, which is either $3$ or $4$ .

Keywords

dynamical degreespolynomial endomorphismsmonomial valuations

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 12 , December 2022 , pp. 3551 - 3592
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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