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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2018
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1812.00057 |
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| _version_ | 1866910212410048512 |
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| author | Espitia, Marcielis Ponce, Gabriel Varão, Régis |
| author_facet | Espitia, Marcielis Ponce, Gabriel Varão, Régis |
| contents | Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $μ$ defined on the completion of the Borel $σ$-algebra and $\mathcal F$ a $f$-invariant one dimensional continuous foliation of $M$ by $C^1$-leaves. Then, if $f$ preserves a continuous $\mathcal{F}$-arc length system, then we only have three possibilities for the conditional measures of $μ$ along $\mathcal F$, namely: - they are atomic for almost every leaf, or - for almost every leaf they are equivalent to the measure $λ_x$ induced by the invariant arc-length system over $\mathcal F$, or - for almost every leaf their support is a nowhere dense, perfect subset of the leaf. Furthermore, we show that restricted to ergodic partially hyperbolic diffeomorphism with one-dimensional topological neutral center direction, we are able to eliminate the third case obtaining a dichotomy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1812_00057 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Classification of Conditional Measures Along Certain Invariant One-Dimensional Foliations Espitia, Marcielis Ponce, Gabriel Varão, Régis Dynamical Systems Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $μ$ defined on the completion of the Borel $σ$-algebra and $\mathcal F$ a $f$-invariant one dimensional continuous foliation of $M$ by $C^1$-leaves. Then, if $f$ preserves a continuous $\mathcal{F}$-arc length system, then we only have three possibilities for the conditional measures of $μ$ along $\mathcal F$, namely: - they are atomic for almost every leaf, or - for almost every leaf they are equivalent to the measure $λ_x$ induced by the invariant arc-length system over $\mathcal F$, or - for almost every leaf their support is a nowhere dense, perfect subset of the leaf. Furthermore, we show that restricted to ergodic partially hyperbolic diffeomorphism with one-dimensional topological neutral center direction, we are able to eliminate the third case obtaining a dichotomy. |
| title | Classification of Conditional Measures Along Certain Invariant One-Dimensional Foliations |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/1812.00057 |