Equidistribution for meromorphic maps with dominant topological degree (Q2788639)

The article under review is concerned with the dynamics of dominant meromorphic self-maps \(f:X\rightarrow X\) of a compact Kähler manifold \(X\) of dimension \(k\geq1\). Such a map induces a linear action on \(H^{\ell,\ell}(X,\mathbb{R})\), and its dynamical degree of order \(\ell\) is the spectral radius \(d_\ell(f)\) of this linear map. The authors assume here that \(f\) has large topological degree, i.e., that the topological degree \(d_k(f)\) satisfies \(d_k(f)>\max_{\ell\neq k}d_\ell(f)\).NEWLINENEWLINEUnder the above assumptions, the map \(f\) admits a unique measure \(\mu\) of maximal entropy \(\log d_k(f)\). The authors prove here that NEWLINE\[NEWLINE\frac{1}{d_k(f)^n}\sum_{z\in \mathcal{Q}_n}\delta_z\longrightarrow\mu,NEWLINE\]NEWLINE as \(n\rightarrow\infty\), where \(\mathcal{Q}_n\) is the set of isolated periodic (resp. repelling periodic) points of period \(n\) and where \(\delta_z\) denotes the Dirac mass at \(z\). The authors first prove the equidistribution of preimages of points outside a pluripolar set (bigger than the indeterminacy set \(I(f)\) of \(f\)). More precisely, let \(I_\infty\cup I_\infty'\) be the pluripolar set of points \(a\in X\) for which \(f^n(a)\in I(f)\), or \(f^{-n}\{a\}\) has a positive dimensional irreducible component for some \(n\geq0\). This set is pluripolar and the authors prove that there exists a (possibly empty) proper analytic set \(\mathcal{E}\subset X\) such that, for all \(a\in X\setminus (I_\infty\cup I_\infty')\), NEWLINE\[NEWLINEd_k(f)^{-n}(f^n)^*\delta_a\rightarrow\mu \text{ as }n\rightarrow\infty \text{ if and only if } a\notin \mathcal{E}.NEWLINE\]NEWLINE This result is obtained using the strategy of the first author and \textit{N. Sibony} in the case of polynomial-like maps [J. Math. Pures Appl. (9) 82, No. 4, 367--423 (2003; Zbl 1033.37023)]. Once this is proved, the key ingredient is the recent theory of density of currents established by \textit{N. Sibony} and the first author [``Density of positive closed currents, a theory of non-generic intersections'', Preprint, \url{arXiv:1203.5810}]: this is used to obtain an upper bound on the number of points: NEWLINE\[NEWLINE\sharp\mathcal{Q}_n\leq d_k(f)^n+o(d_k(f)^n).NEWLINE\]NEWLINE Once this is proved, the authors follow the proof used by \textit{J.-Y. Briend} and \textit{J. Duval} [Publ. Math., Inst. Hautes Étud. Sci. 93, 145--159 (2001; Zbl 1010.37004)] in the case of holomorphic endomorphisms of projective spaces.