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Holonomies for foliations with extreme disintegration behavior

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Abstract

Given a foliation, we say that it displays “extreme disintegration behavior” if either it has atomic disintegration or if it is leafwise absolutely continuous and its conditional measures are uniformly equivalent to the leaf volume, which we call UDB property. Both concepts are related to the decomposition of volume with respect to the foliation. We relate these behaviors to the measure-theoretical regularity of the holonomies, by proving that a foliation with atomic disintegration has holonomies taking full volume sets to zero volume sets, and we characterize the UBD property with the holonomies having uniformly bounded Jacobians. Both extreme phenomena appear in invariant foliations for dynamical systems.

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... In recent work, both authors and M. Cantarino [5] have tried to better understand atomic disintegration in terms of the holonomy map. It is proved in [5] that if a foliation has atomic disintegration with respect to the Lebesgue measure then for almost every pair of transversal the holonomy takes a set of full Lebesgue measure to a set of zero Lebesgue measure. ...
... In recent work, both authors and M. Cantarino [5] have tried to better understand atomic disintegration in terms of the holonomy map. It is proved in [5] that if a foliation has atomic disintegration with respect to the Lebesgue measure then for almost every pair of transversal the holonomy takes a set of full Lebesgue measure to a set of zero Lebesgue measure. The holonomy gives an idea of how each leaf of the foliation is stacked on one another. ...
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