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Geometric and arithmetic properties of the Hénon map. (English) Zbl 0807.58021
The paper studies some of the algebro-geometric and number theoretic properties of the Hénon map \(\phi(x,t) = (y,y^ 2 + b + ax)\). This map is extended to a map on the projective plane \(P^ 2\). The theory of height-functions on \(P^ 2\) is a tool for a description of the properties of the Hénon map.
The main results are the following. Let \(\phi\) be a Hénon map defined over \(\overline{Q}\) (apparently, the author means \(\overline{Q}\) as the rational numbers plus infinity). a) The set of periodic points is a set of bounded height. In particular, if \(\phi\) is defined over a number field \(K\) then there are only finitely many periodic points of \(\phi\) with coordinates in \(K\). b) If \(p\) is a non-periodic point and \(h\) the usual height function then \(\log h(\phi^ n p) \approx | n| \log 2\) as \(n \to \pm \infty\).
Some numerical examples of Hénon maps with rational periodic points are given.

37B99 Topological dynamics
14E05 Rational and birational maps
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