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 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.
Inspired by the works of Zagier, we study the probability measures $ u(t)$ with support on the flat tori which are the compact orbits of the maximal unipotent subgroup acting holomorphically on the positive orthonormal frame bundle $mathcal{F}({M}_D)$ of 3-dimensional hyperbolic Bianchi orbifolds ${M}_D=mathbb{H}^3/widetilde{Gamma}_D$, of finite volume and with only one cusp. Here $Gamma_D=PSL(2, mathcal{O})$, where $mathcal{O}$ is the ring of integers of an imaginary quadratic field of class number one.
We describe several methods to construct minimal foliations by hyperbolic surfaces on closed 3-manifolds, and discuss the properties of the examples thus obtained.
Shigenori Matsumoto
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(2014)
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"Weak form of equidistribution theorem for harmonic measures of foliations by hyperbolic surfaces"
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Shigenori Matsumoto
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