Simeão Targino da Silva’s research while affiliated with Federal Rural University of Semi-Árido and other places

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Publications (2)


Holonomies for foliations with extreme disintegration behavior
  • Article

July 2024

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7 Reads

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2 Citations

Stochastics and Dynamics

Marisa Cantarino

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Simeão Targino da Silva

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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.


Conditions for Atomic Disintegration to be Monoatomic
  • Article
  • Publisher preview available

September 2023

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20 Reads

Journal of Dynamical and Control Systems

In this work, we consider conditions in which atomic disintegration is in fact a monoatomic disintegration. One of the motivations of this work comes from the work of Ponce et al. (J Mod Dyn, 8(1):93–107, 2014); they prove that there is a minimal foliation and a set of full volume which intersects each leaf in one point, but their argument uses some of the hyperbolic structure of the system. We generalize some of their techniques in which we eliminate the need for Markov partitions, which are structures inherited from Anosov diffeomorphisms. We prove some results on which atomic disintegration and some contraction hypothesis on the foliation implies monoatomic disintegration.

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Citations (1)


... 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. ...

Reference:

Conditions for Atomic Disintegration to be Monoatomic
Holonomies for foliations with extreme disintegration behavior
  • Citing Article
  • July 2024

Stochastics and Dynamics