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Gromov's compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov's compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin's representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry.

No abstract uploaded!

No abstract uploaded!

We prove rigidity results for a class of non-uniformly hyperbolic holomorphic maps. If a holomorphic Collet-Eckmann map$f$is topologically conjugate to a holomorphic map$g$, then the conjugacy can be improved to be quasiconformal. If there is only one critical point in the repeller, then$g$is Collet-Eckmann, too.