Foliated hyperbolicity and foliations with hyperbolic leaves
International audience Given a lamination in a compact space and a laminated vector field $X$ which is hyperbolic when restricted to the leaves of the lamination, we distinguish a class of $X$-invariant probabilities that describe the behavior of almost every $X$-orbit in every leaf, which we call G...
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Online Access: | https://hal.archives-ouvertes.fr/hal-01996909 https://doi.org/10.1017/etds.2018.61 |
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ftccsdartic:oai:HAL:hal-01996909v1 2023-05-15T18:23:01+02:00 Foliated hyperbolicity and foliations with hyperbolic leaves Bonatti, Christian GÓMEZ-MONT, XAVIER Martínez, Matilde Institut de Mathématiques de Bourgogne Dijon (IMB) Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC) Université Bourgogne Franche-Comté COMUE (UBFC)-Université Bourgogne Franche-Comté COMUE (UBFC)-Université de Bourgogne (UB) Centro de Investigación en Matemáticas (CIMAT) Consejo Nacional de Ciencia y Tecnología Mexico (CONACYT) Universidad de la República Montevideo (UCUR) Centre National de la Recherche Scientifique (CNRS) ECOS Nord Conacyt 249542Consejo Nacional de Ciencia y Tecnologia (CONACyT)134081LAISLA CIMAT ANII-FCE 2007-106 2020-04 https://hal.archives-ouvertes.fr/hal-01996909 https://doi.org/10.1017/etds.2018.61 en eng HAL CCSD Cambridge University Press (CUP) info:eu-repo/semantics/altIdentifier/arxiv/1510.05026 info:eu-repo/semantics/altIdentifier/doi/10.1017/etds.2018.61 hal-01996909 https://hal.archives-ouvertes.fr/hal-01996909 ARXIV: 1510.05026 doi:10.1017/etds.2018.61 ISSN: 0143-3857 EISSN: 1469-4417 Ergodic Theory and Dynamical Systems https://hal.archives-ouvertes.fr/hal-01996909 Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2020, 40 (4), pp.881-903. ⟨10.1017/etds.2018.61⟩ https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/foliated-hyperbolicity-and-foliations-with-hyperbolic-leaves/19233D57850504820D7A8AD46B59FBA4# geodesic-flow harmonic-measures [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] info:eu-repo/semantics/article Journal articles 2020 ftccsdartic https://doi.org/10.1017/etds.2018.61 2021-11-07T02:19:39Z International audience Given a lamination in a compact space and a laminated vector field $X$ which is hyperbolic when restricted to the leaves of the lamination, we distinguish a class of $X$-invariant probabilities that describe the behavior of almost every $X$-orbit in every leaf, which we call Gibbs $u$-states. We apply this to the case of foliations in compact manifolds having leaves with negative curvature, using the foliated hyperbolic vector field on the unit tangent bundle to the foliation generating the leaf geodesics. When the Lyapunov exponents of such ergodic Gibbs $u$-states are negative, it is an SRB measure (having a positive Lebesgue basin of attraction). When the foliation is by hyperbolic leaves, this class of probabilities coincide with the classical harmonic measures introduced by Garnett. Furthermore, if the foliation is transversally conformal and does not admit a transverse invariant measure we show that there are finitely many ergodic Gibbs $u$-states, each supported in one minimal set of the foliation, each having negative Lyapunov exponents, and the union of their basins of attraction has full Lebesgue measure. The leaf geodesics emanating from a point have a proportion whose asymptotic statistics are described by each of these ergodic Gibbs -states, giving rise to continuous visibility functions of the attractors. Reversing time, by considering $-X$ , we obtain the existence of the same number of repellers of the foliated geodesic flow having the same harmonic measures as projections to $M$. In the case of only one attractor, we obtain a north to south pole dynamics. Article in Journal/Newspaper South pole Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) South Pole Ergodic Theory and Dynamical Systems 40 4 881 903 |
institution |
Open Polar |
collection |
Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
op_collection_id |
ftccsdartic |
language |
English |
topic |
geodesic-flow harmonic-measures [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] |
spellingShingle |
geodesic-flow harmonic-measures [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] Bonatti, Christian GÓMEZ-MONT, XAVIER Martínez, Matilde Foliated hyperbolicity and foliations with hyperbolic leaves |
topic_facet |
geodesic-flow harmonic-measures [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] |
description |
International audience Given a lamination in a compact space and a laminated vector field $X$ which is hyperbolic when restricted to the leaves of the lamination, we distinguish a class of $X$-invariant probabilities that describe the behavior of almost every $X$-orbit in every leaf, which we call Gibbs $u$-states. We apply this to the case of foliations in compact manifolds having leaves with negative curvature, using the foliated hyperbolic vector field on the unit tangent bundle to the foliation generating the leaf geodesics. When the Lyapunov exponents of such ergodic Gibbs $u$-states are negative, it is an SRB measure (having a positive Lebesgue basin of attraction). When the foliation is by hyperbolic leaves, this class of probabilities coincide with the classical harmonic measures introduced by Garnett. Furthermore, if the foliation is transversally conformal and does not admit a transverse invariant measure we show that there are finitely many ergodic Gibbs $u$-states, each supported in one minimal set of the foliation, each having negative Lyapunov exponents, and the union of their basins of attraction has full Lebesgue measure. The leaf geodesics emanating from a point have a proportion whose asymptotic statistics are described by each of these ergodic Gibbs -states, giving rise to continuous visibility functions of the attractors. Reversing time, by considering $-X$ , we obtain the existence of the same number of repellers of the foliated geodesic flow having the same harmonic measures as projections to $M$. In the case of only one attractor, we obtain a north to south pole dynamics. |
author2 |
Institut de Mathématiques de Bourgogne Dijon (IMB) Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC) Université Bourgogne Franche-Comté COMUE (UBFC)-Université Bourgogne Franche-Comté COMUE (UBFC)-Université de Bourgogne (UB) Centro de Investigación en Matemáticas (CIMAT) Consejo Nacional de Ciencia y Tecnología Mexico (CONACYT) Universidad de la República Montevideo (UCUR) Centre National de la Recherche Scientifique (CNRS) ECOS Nord Conacyt 249542Consejo Nacional de Ciencia y Tecnologia (CONACyT)134081LAISLA CIMAT ANII-FCE 2007-106 |
format |
Article in Journal/Newspaper |
author |
Bonatti, Christian GÓMEZ-MONT, XAVIER Martínez, Matilde |
author_facet |
Bonatti, Christian GÓMEZ-MONT, XAVIER Martínez, Matilde |
author_sort |
Bonatti, Christian |
title |
Foliated hyperbolicity and foliations with hyperbolic leaves |
title_short |
Foliated hyperbolicity and foliations with hyperbolic leaves |
title_full |
Foliated hyperbolicity and foliations with hyperbolic leaves |
title_fullStr |
Foliated hyperbolicity and foliations with hyperbolic leaves |
title_full_unstemmed |
Foliated hyperbolicity and foliations with hyperbolic leaves |
title_sort |
foliated hyperbolicity and foliations with hyperbolic leaves |
publisher |
HAL CCSD |
publishDate |
2020 |
url |
https://hal.archives-ouvertes.fr/hal-01996909 https://doi.org/10.1017/etds.2018.61 |
geographic |
South Pole |
geographic_facet |
South Pole |
genre |
South pole |
genre_facet |
South pole |
op_source |
ISSN: 0143-3857 EISSN: 1469-4417 Ergodic Theory and Dynamical Systems https://hal.archives-ouvertes.fr/hal-01996909 Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2020, 40 (4), pp.881-903. ⟨10.1017/etds.2018.61⟩ https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/foliated-hyperbolicity-and-foliations-with-hyperbolic-leaves/19233D57850504820D7A8AD46B59FBA4# |
op_relation |
info:eu-repo/semantics/altIdentifier/arxiv/1510.05026 info:eu-repo/semantics/altIdentifier/doi/10.1017/etds.2018.61 hal-01996909 https://hal.archives-ouvertes.fr/hal-01996909 ARXIV: 1510.05026 doi:10.1017/etds.2018.61 |
op_doi |
https://doi.org/10.1017/etds.2018.61 |
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Ergodic Theory and Dynamical Systems |
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40 |
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881 |
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