Ergodic properties of rational mappings with large topological degree
Ergodic properties of rational mappings with large topological degree
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大拓扑度有理映射的遍历性质
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发表时间：
2005
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通讯作者：
V. Guedj
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作者：
V. Guedj
设X是射影流形，f：X？？X是拓扑度大的有理映射，dt >？Ek.1（f）：=（k . 1）f的第n动力学度。我们给出一个概率测度的初等构造？Ef such that d.n t（fn）.？？?? ? Ef对于每个光滑概率测度？？在X上。我们证明了每个拟多重次调和函数是？Ef -可积。特别是？EF不收取任何点的不确定性或多极集，因此？Ef是f不变的，雅可比f是常数。Ef = dt？EF .然后，我们建立的主要遍历性质？EF：它是混合与积极的李雅普诺夫指数，原像？hmost？h点以及排斥周期点是均匀分布的关于？EF .此外，当dimC X . 3或当X是复齐次的，？Ef是最大熵的唯一度量。
Let X be a projective manifold and f : X ?? X a rational mapping with large topological degree, dt > ?Ek.1(f) := the (k . 1)th dynamical degree of f. We give an elementary construction of a probability measure ?Ef such that d.n t (fn).?? ?? ?Ef for every smooth probability measure ?? on X. We show that every quasiplurisubharmonic function is ?Ef -integrable. In particular ?Ef does not charge either points of indeterminacy or pluripolar sets, hence ?Ef is f-invariant with constant jacobian f.?Ef = dt?Ef . We then establish the main ergodic properties of ?Ef : it is mixing with positive Lyapunov exponents, preimages of ?hmost?h points as well as repelling periodic points are equidistributed with respect to ?Ef . Moreover, when dimC X . 3 or when X is complex homogeneous, ?Ef is the unique measure of maximal entropy.