Dynamics of bimeromorphic maps of surfaces

We classify bimeromorphic self-maps f: X ○ of compact Kähler surfaces X in terms of their actions f*: H^1,1(X) ○ 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 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.