The paper is devoted to the question of the existence of compositional roots for the Hénon map \(H(z,w)=(w,p(w)-az)\), where \(p\) is a monic polynomial of degree \(d\geq 2\), \(a\in \mathbb{C}\setminus 0\}\). Let \(\mathcal C\) denote the space of finite compositions of such maps. The authors prove that any root of a map in \(\mathcal C\) must be a polynomial map and any map in \(\mathcal C\) can have roots of only a finite number of distinct orders. Interesting observations of the authors also include the fact that, for the remaining elementary maps which are not the time-1 map of a flow, they can prove that such maps have roots of arbitrarily high order and nonpolynomial roots, but any root of such a map is conjugate to a polynomial elementary map.
For the entire collection see [Zbl 0903.00037].