Dynamics of bimeromorphic maps of surfaces

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 *: H 1,1 ( X ) [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i "/] on cohomology. We observe that the growth rate of ║ f n *║ is invariant under bimeromorphic conjugacy, and that by conjugating one can always arrange that f n * = f * n . We show that the sequence ║ f n *║ 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 θ 2 + = 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.