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 $\mathbb{A}^r$?



If the sequence of degrees of the iterates of a self-map of $\mathbb{A}^2$ is bounded, is it eventually periodic?
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 $\mathbb C^2$"](http://arxiv.org/abs/0711.2770).


