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Register a new userHorocycle flows for laminations by hyperbolic Riemann surfaces and Hedlunds theorem
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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.
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We show that the horocycle flow associated with a foliation on a compact manifold by hyperbolic surfaces is minimal under certain conditions.
We show that the horocycle flows of open tight hyperbolic surfaces do not admit minimal sets.
We show that the equidistribution theorem of C. Bonatti and X. Gomez-Mont for a special kind of foliations by hyperbolic surfaces does not hold in general, and seek for a weaker form valid for general foliations by hyperbolic surfaces.
We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured Riemann surface laminations induced by Levy-Prokhorov metric. As an application of the compactness theorem, we show that given a biholomorphism of $phi $ of a closed complex manifold $X$, some power $phi^k $ ($k>0$) fixes a measured Riemann surface lamination in $X$.
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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