Dynamics of bimeromorphic maps of surfaces.

*(English)*Zbl 1112.37308Summary: We classify bimeromorphic self-maps \(f \colon \;X \circlearrowleft \) of compact Kähler surfaces \(X\) in terms of their actions \(f^* \colon \;H^{1,1}(X) \circlearrowleft \) 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 \(\theta_+\) for \(f^*\), that \(\theta_+\) is nef, and that \(f\) is bimeromorphically conjugate to an automorphism if and only if \(\theta_+^2 = 0\). We go on in this case to construct a dynamically natural positive current representing \(\theta_+\), and we study the growth rate of periodic orbits of \(f\). We conclude by illustrating our results with a particular family of examples.