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Let $f : \mathbb{A}_k^2 \to \mathbb{A}_k^2$ be a regular self-map of the affine plane over a field $k$ of characteristic zero. Assume that the sequence $(\deg{f^n})_{n \in \mathbb{N}}$ is bounded. Is this sequence eventually periodic?

[An embarrassing question... ]

Added: The Favre-Jonnson paper linked to in Gjergji Zaimi's answer settles the question for the case of a self-map of the affine plane. How about the generalization with $\mathbb{A}^2$ replaced with $\mathbb{A}^r$?

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  • $\begingroup$ Do I understand the question correctly? Any (non-trivial) translation seems to be a counter-example... $\endgroup$
    – abx
    Commented Dec 1, 2013 at 9:38
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    $\begingroup$ I ask about the sequence of degrees being eventually periodic - not about the map itself. $\endgroup$
    – Vesselin Dimitrov
    Commented Dec 1, 2013 at 9:38

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It's a theorem of Favre and Jonsson that the degree sequence satisfies a linear recurrence, so your claim follows. See their paper "Dynamical compactifications of $\mathbb C^2$".

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  • $\begingroup$ Thank you -- I was not aware of this. This paper is very particular to self-maps of the affine plane. How about the same question for a rational iteration $\mathbb{P}^r \dashrightarrow \mathbb{P}^r$? Then the power series $\sum_{n \geq 0} \deg{(f^n)}T^n$ need not be rational; any general result available in this direction? $\endgroup$
    – Vesselin Dimitrov
    Commented Dec 1, 2013 at 10:29

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