Abstract
The polynomial automorphisms of the affine plane have been studied a lot: if f is such an automorphism, then either f preserves a rational fibration, has an uncountable centralizer, and its first dynamical degree equals 1, or f preserves no rational curves, has a countable centralizer, and its first dynamical degree is >1. In higher dimensions there is no such description. In this article we study a family \((\Psi _{ \upalpha })_{ \upalpha }\) of polynomial automorphisms of \(\mathbb {C}^3\). We show that the first dynamical degree of \({\mathbf \Psi }_{ \upalpha }\) is \(>1\), that \({\Psi }_{ \upalpha }\) preserves a unique rational fibration and has an uncountable centralizer. We then describe the dynamics of the family \(({\Psi }_{ \upalpha })_{ \upalpha }\), in particular the speed of points escaping to infinity. We also observe different behaviors according to the value of the parameter \({ \upalpha }\).
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Notes
Indeed if \(p=(p_0,p_1,p_2)\) satisfies \(p_2 \ne 0\) and \(p_1 = - {\upvarphi } p_0\), we see that \(P_{{\upalpha }}^{(0)}(p) + {\upvarphi } P_{{\upalpha }}^{(1)}(p)=p_0 (1 + {\upvarphi } - {\upvarphi }^2) + {\upvarphi } p_0^q p_2^d = {\upvarphi } p_0^q p_2^d \ne 0\) hence \(\Delta _{{\upvarphi }'} \times \mathbb {C}\) is not left invariant by \(\Psi _{{\upalpha }}\).
In particular, this is satisfied for points \(p \in \Omega \) as we have seen in Lemma 8.3.
Indeed, if \(p \in W_{\Psi _{{\upalpha }}}^s(0_{\mathbb {C}^3})\), then \(h(p) \in W_{{\upphi }_{{\upalpha }}}^s(0_{\mathbb {C}^2})\); conversely, if \((p_0,p_1) \in K_{{\upphi }_{{\upalpha }}}^+\), then \((p_0,p_1)=h(p_0,p_1,1)\) and \((p_0,p_1,1) \in K_{\Psi _{{\upalpha }}}^+=W_{\Psi _{{\upalpha }}}^s(0_{\mathbb {C}^3})\).
Else there exists \(n_1<n_0\) such that \(|P_{{\upalpha }}^{(n_1)}(p)| > ((1-{\upvarepsilon }){\upvarphi })^{n_0}\) and \(|P_{{\upalpha }}^{(n_1)}(p)| \ge |P_{{\upalpha }}^{(n_1-1)}(p)|\) and we consider \(n_1\) instead of \(n_0\).
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The first author would like to thank Artur Avila for helpful and fruitful discussions.
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Déserti, J., Leguil, M. Dynamics of a Family of Polynomial Automorphisms of \(\mathbb {C}^3\), a Phase Transition. J Geom Anal 28, 190–224 (2018). https://doi.org/10.1007/s12220-017-9816-1
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DOI: https://doi.org/10.1007/s12220-017-9816-1