Abstract

We classify bimeromorphic self-maps f: X [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] of compact Kähler surfaces X in terms of their actions f*: H1,1(X) [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i "/] on cohomology. We observe that the growth rate of ║fn*║ is invariant under bimeromorphic conjugacy, and that by conjugating one can always arrange that fn* = f*n. We show that the sequence ║fn*║ can be bounded, grow linearly, grow quadratically, or grow exponentially. In the first three cases, we show that after conjugating, f is an automorphism virtually isotopic to the identity, f preserves a rational fibration, or f preserves an elliptic fibration, respectively. In the last case, we show that there is a unique (up to scaling) expanding eigenvector θ+ for f*, that θ+ is nef, and that f is bimeromorphically conjugate to an automorphism if and only if θ2+ = 0. We go on in this case to construct a dynamically natural positive current representing θ+, and we study the growth rate of periodic orbits of f. We conclude by illustrating our results with a particular family of examples.

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