Ergodic properties of rational mappings with large topological degree

• 成果类型: Article
• 署名作者: Guedj, V
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2005.161.1589
• 发表日期: 2005
• 页码: 1589-1607
• 关键词: dynamics; entropy; points; MAPS

摘要

Let X be a projective manifold and f : X -> X a. rational mapping with large topological degree, d(t) > lambda(k-1)(f) := the (k - 1)(th) dynamical degree of f. We give an elementary construction of a probability measure mu(f) such that d(t)(-n)(f(n))*Theta -> mu(f) for every smooth probability measure Theta on X. We show that every quasiplurisubharmonic function is mu(f)-integrable. In particular mu(f) does not charge either points of indeterminacy or pluripolar sets, hence mu(f) is f-invariant with constant jacobian f*mu(f) = d(t)mu(f). We then establish the main ergodic properties of mu(f): it is mixing with positive Lyapunov exponents, preimages of most points as well as repelling periodic points are equidistributed with respect to mu(f). Moreover, when dim(C) X <= 3 or when X is complex homogeneous, mu(f) is the unique measure of maximal entropy.