Ergodic properties of rational mappings with large topological degree | Annals of Mathematics

Abstract Let X be a projective manifold and f:X→X a rational mapping with large topological degree, dt>λk−1(f):= the (k−1)th dynamical degree of f. We give an elementary construction of a probability measure μf such that d−nt(fn)∗Θ→μf for every smooth probability measure Θ on X. We show that every quasiplurisubharmonic function is μf-integrable. In particular μf does not charge either points of indeterminacy or pluripolar sets, hence μf is f-invariant with constant jacobian f∗μf=dtμf. We then establish the main ergodic properties of μf: it is mixing with positive Lyapunov exponents, preimages of “most” points as well as repelling periodic points are equidistributed with respect to μf. Moreover, when dimℂX≤3 or when X is complex homogeneous, μf is the unique measure of maximal entropy.

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数学年刊(Annals of Mathematics)

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