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Conditions for Atomic Disintegration to be Monoatomic

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Abstract

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.
Journal of Dynamical and Control Systems (2023) 29:1943–1960
https://doi.org/10.1007/s10883-023-09665-x
Conditions for Atomic Disintegration to be Monoatomic
Simeão Targino da Silva1·Régis Varão2
Received: 7 November 2022 / Revised: 7 November 2022 / Accepted: 2 September 2023 /
Published online: 22 September 2023
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023
Abstract
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.
Keywords Ergodic theory ·Measure disintegration ·Nonuniformly hyperbolic systems
Mathematics Subject Classification (2010): 28A50 ·37A05 ·37D25
1 Introduction
An essential class of dynamic systems that have been widely studied due to their rich dynam-
ical properties is the Anosov diffeomorphisms. Associated with these diffeomorphisms are
the geometric structures known as stable and unstable foliations. A fundamental result of the
hyperbolic theory, proved by Anosov [1], is that any Anosov diffeomorphism of class C2that
preserves volume is ergodic. One of the fundamental points of this result is understanding the
metric behavior of these foliations (stable and unstable). More precisely, understand that the
disintegration of the volume concerning the stable and unstable foliations provides measures
equivalent to the Lebesgue measure of the leaf, that is, the stable and unstable foliations
behave regularly enough so that one can “apply” Fubini’s theorem.
The metric behavior of foliations associated with a dynamic is the key point of this work.
The behavior of a foliation could be understood as how it disintegrates the measures on the
leaves. Given a probability space (M,B),whereMis a compact manifold and Bis its
Borel σ-algebra, a disintegration of a measure μis a family {μP:PP}of probabilities
BRégis Varão
varao@unicamp.br
1Department of Natural Sciences, Mathematics and Statistics, UFERSA, Mossoró, Brazil
2Institute of Mathematics, Statistics and Scientific Computation, UNICAMP, São Paulo, Brazil
123
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A global geometric and probabilistic perspective, Mathematical physics III
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