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Local and global canonical height functions for affine space regular automorphisms. (English) Zbl 1302.37067

Summary: Let \(f:\mathbb A^N\to \mathbb A^N\) be a regular polynomial automorphism defined over a number field \(K\). For each place \(v\) of \(K\), we construct the \(v\)-adic Green functions \(G_{f,v}\) and \(G_{f^{-1},v}\) (i.e., the \(v\)-adic canonical height functions) for \(f\) and \(f^{-1}\). Next we introduce for \(f\) the notion of good reduction at \(v\), and using this notion, we show that the sum of \(v\)-adic Green functions over all \(v\) gives rise to a canonical height function for \(f\) that satisfies a Northcott-type finiteness property. Using an earlier result, we recover results on arithmetic properties of \(f\)-periodic points and non-\(f\)-periodic points. We also obtain an estimate of growth of heights under \(f\) and \(f^{-1}\), which was independently obtained by C. G. Lee [Math. Ann. 355, No. 1, 1–16 (2013; Zbl 1330.11045)] by a different method.

MSC:

37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
11G50 Heights
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P20 Dynamical systems over non-Archimedean local ground fields

Citations:

Zbl 1330.11045
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