Canonical height functions for affine plane automorphisms (Q2494341)

There are several papers which are concerned with the construction of a canonical height on a given projective variety, that behaves well with respect to a given morphism or a given class of morphisms, see for instance \textit{J. H. Silverman} [Invent. Math. 105, 347--373 (1991; Zbl 0754.14023)], \textit{G. S. Call} and \textit{J. H. Silverman} [Compos. Math. 89, 163--205 (1993; Zbl 0826.14015)] and the author [J. Reine Angew. Math. 597, 135--174 (2006; Zbl 1109.14025)]. In the present paper, the author constructs canonical heights on the affine plane, behaving well with respect to a given polynomial automorphism of that plane. Recall that a polynomial automorphism of the affine plane \({\mathbf A}^2\) defined over a number field \(K\) is an invertible map \(f: {\mathbf A}^2\to {\mathbf A}^2: (x,y)\mapsto (p(x,y),q(x,y))\), where \(p,q\) are polynomials in \(K[x,y]\). The degree of such a map \(f\) is given by \(\deg f:=\max (\deg p,\deg q)\), and the dynamical degree of \(f\) by \(\delta (f):=\lim_{n\to\infty} (\deg f^n)^{1/n}\), where \(f^n\) is the \(n\)-th iterate of \(f\). It is known that \(\delta (f)\) is an integer with \(1\leq \delta (f)\leq \deg (f)\). The author proves among other things the following. Let \(K\) be an algebraic number field with algebraic closure \(\overline{K}\) and denote by \(h\) the naive logarithmic height on \({\mathbf A}^2(\overline{K})\). Let \(f\) be a polynomial automorphism of \({\mathbf A}^2\) defined over \(K\) of dynamical degree \(\delta \geq 2\). Then there exists a function \(\widehat{h}:{\mathbf A}^2(\overline{K})\) (called canonical height function for \(f\)) with the following properties: (i) there are constants \(a_1,a_2>0\) and \(b_1,b_2\) such that \(a_1h(x)+b_1\leq \widehat{h}(x)\leq a_2h(x)+b_2\) for \(x\in {\mathbf A}^2(\overline{K})\); (ii) \(\widehat{h}(f(x))+\widehat{h}(f^{-1}(x))=(\delta +\delta^{-1})\widehat{h}(x)\) for \(x\in {\mathbf A}^2(\overline{K})\). Further, any two canonical height functions for \(f\) that differ by a bounded function on \({\mathbf A}^2(\overline{K})\) must be equal. Given \(x\in {\mathbf A}^2(\overline{K})\), denote by \(O_f(x)\) the orbit of \(x\) with respect to \(f\), i.e.,\(\{ f^n(x): n\in\mathbb{Z}\}\). The author deduces that \(\widehat{h}(x)\geq 0\) for \(x\in {\mathbf A}^2(\overline{K})\) and \(f(x)=0\) if and only if \(O_f(x)\) is finite. In case that \(O_f(x)\) is not finite, the author proves an asymptotic formula as \(T\to\infty\) for the number of points \(y\in O_f(x)\) of height \(h(y)\leq T\) with main term \({2\over\log\delta}\log T\).