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Register a new userIterated convolutions and endless Riemann surfaces
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We discuss a version of Ecalles definition of resurgence, based on the notion of endless continuability in the Borel plane. We relate this with the notion of Omega-continuability, where Omega is a discrete filtered set, and show how to construct a universal Riemann surface X_Omega whose holomorphic functions are in one-to-one correspondence with Omega-continuable functions. We then discuss the Omega-continuability of convolution products and give estimates for iterated convolutions of the form hatphi_1*cdots *hatphi_n. This allows us to handle nonlinear operations with resurgent series, e.g. substitution into a convergent power series.
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We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(hat M, T^1mathcal{F})$ of a compact minimal lamination $(M,mathcal F)$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal, and examples where this action is not minimal. In the first case, we prove that if $mathcal F$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.
For a generic set in the Teichmueller space, we construct a covariant functor with the range in a category of the AF-algebras; the functor maps isomorphic Riemann surfaces to the stably isomorphic AF-algebras. As a special case, one gets a categorical correspondence between complex tori and the so-called Effros-Shen algebras.
We construct a converging geometric iterated function system on the moduli space of ordered triangles, for which the involved functions have geometric meanings and contain a non-contraction map under the natural metric.
We develop linear discretization of complex analysis, originally introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We prove convergence of discrete period matrices and discrete Abelian integrals to their continuous counterparts. We also prove a discrete counterpart of the Riemann--Roch theorem. The proofs use energy estimates inspired by electrical networks.
The Quillen connection on ${mathcal L} rightarrow {mathcal M}_g$, where ${mathcal L}^*$ is the Hodge line bundle over the moduli stack of smooth complex projective curves curves ${mathcal M}_g$, $g geq 5$, is uniquely determined by the condition that its curvature is the Weil--Petersson form on ${mathcal M}_g$. The bundle of holomorphic connections on ${mathcal L}$ has a unique holomorphic isomorphism with the bundle on ${mathcal M}_g$ given by the moduli stack of projective structures. This isomorphism takes the $C^infty$ section of the first bundle given by the Quillen connection on ${mathcal L}$ to the $C^infty$ section of the second bundle given by the uniformization theorem. Therefore, any one of these two sections determines the other uniquely.
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