Equidistribution of periodic points of some automorphisms on K3 surfaces (Q2880470)

The aim of this paper under review is to establish an equidistribution theorem for the dynamics arising from a finite set of endomorphisms on a projective variety \(W\). The idea is to combine \textit{X. Yuan}'s general equidistribution theorem [Invent. Math. 173 No. 3, 603--649 (2008; Zbl 1146.14016)] and \textit{S. Kawaguchi}'s result [J. Reine Angew. Math. 597, 135--173 (2006; Zbl 1109.14025)] on the canonical height associated to a finite set of morphisms on \(W\).NEWLINENEWLINETo state the main result of this paper, let \(W\) be a projective variety of dimension \(n\) defined over a number field \(K\) and let \(M = \{\phi_1, \ldots, \phi_t : W \to W\}\) be a finite set of endomorphisms on \(W\). The dynamical system \((W,M)\) is said to be polarizable if there is an ample line bundle \(L\in \mathrm{Div}(W)\otimes {\mathbb Q}\) such that \(\otimes_{i=1}^t \phi_i^{\ast}L \sim L^{\otimes q}\) for some rational number \(q > t.\) Assume that we're in this situation, then Kawaguchi~[ibid] shows that there is a metric \(\|\cdot\|_M\) on \(L\) such that a unique height function \({\hat h}_{\bar L}\) on \(W({\bar K})\) associated to the metrized line bundle \({\bar L} = (L, \|\cdot\|_M)\) is defined together with some conditions which we omit here. Let \({\mathcal L} = L^{\otimes e}\in \mathrm{Div}(W)\) for some appropriate \(e\in {\mathbb Q}\) . Then, we have that \(\bar{\mathcal L} = (L^{\otimes e}, \|\cdot\|_{M}^e)\) is an adelic semipositive metrized line bundle in the sense of \textit{S. Zhang} [J. Algebr. Geom. 4, No. 2, 281--300; (1995; Zbl 0861.14019)]. Applying Yuan's theorem quoted above, the author concludes that for every place \(v\) of \(K\) there is a probability measure \(\mu_{M,v} = c_1({\mathcal L}^n)/(\deg_L W)\) invariant under the actions of the monoid generated by \(M\) on the analytic space \(W_{K_v}^{an}\) such that any given generic and small sequence of algebraic points \(\{x_m\}\subset W({\bar K})\), the probability measure \(\frac 1{\deg x_m}\sum_{y \in \Gamma_{x_m}} \delta_y\) supported on the Galois orbit \(\Gamma_{x_m}\) of \(x_m\) converges weakly to \(\mu_{M,v}\).NEWLINENEWLINEAnother result of this paper is to show that for the dynamics \(M = \{\phi, \phi^{-1}\}\) generated by automorphisms \(\phi\) of \(W\) such that \((W, M)\) is polarizable, then the set of periodic points \(\mathrm{Per}(\phi)\) is Zariski dense in \(W.\) Combining with the above equidistribution result, the author then concludes that \(\mathrm{Per}(\phi)\) is equidistributed with respect to the probability measure \(\mu_{M,v}\) on \(W_{K_v}^{an}\) for every place \(v\) of \(K\).