## Dynamics of bimeromorphic maps of surfaces.(English)Zbl 1112.37308

Summary: 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.

### MSC:

 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
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