The Experts below are selected from a list of 81 Experts worldwide ranked by ideXlab platform
Ian Melbourne - One of the best experts on this subject based on the ideXlab platform.
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mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth Stable Foliation
Advances in Mathematics, 2019Co-Authors: Vitor Araujo, Ian MelbourneAbstract:Abstract Over the last 10 years or so, advanced statistical properties, including exponential decay of correlations, have been established for certain classes of singular hyperbolic flows in three dimensions. The results apply in particular to the classical Lorenz attractor. However, many of the proofs rely heavily on the smoothness of the Stable Foliation for the flow. In this paper, we show that many statistical properties hold for singular hyperbolic flows with no smoothness assumption on the Stable Foliation. These properties include existence of SRB measures, central limit theorems and associated invariance principles, as well as results on mixing and rates of mixing. The properties hold equally for singular hyperbolic flows in higher dimensions provided the center-unStable subspaces are two-dimensional.
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existence and smoothness of the Stable Foliation for sectional hyperbolic attractors
Bulletin of The London Mathematical Society, 2017Co-Authors: Vitor Araujo, Ian MelbourneAbstract:We prove the existence of a contracting invariant topological Foliation in a full neighbourhood for partially hyperbolic attractors. Under certain bunching conditions, it can then be shown that this Stable Foliation is smooth. Specialising to sectional hyperbolic attractors, we give a verifiable condition for bunching. In particular, we show that the Stable Foliation for the classical Lorenz equation (and nearby vector fields) is better than C1, which is crucial for recent results on exponential decay of correlations. In fact, the Foliation is at least C1.278
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exponential decay of correlations for nonuniformly hyperbolic flows with a c 1 alpha c 1 α Stable Foliation including the classical lorenz attractor
Annales Henri Poincaré, 2016Co-Authors: Vitor Araujo, Ian MelbourneAbstract:We prove exponential decay of correlations for a class of \({C^{1+\alpha}}\) uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open set of geometric Lorenz attractors. As a special case, we show that the classical Lorenz attractor is robustly exponentially mixing.
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exponential decay of correlations for nonuniformly hyperbolic flows with a c 1 alpha Stable Foliation
arXiv: Dynamical Systems, 2015Co-Authors: Vitor Araujo, Ian MelbourneAbstract:We prove exponential decay of correlations for a class of C^{1+\alpha} uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open and dense set of geometric Lorenz attractors in a neighborhood of the classical Lorenz attractor.
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exponential decay of correlations for nonuniformly hyperbolic flows with a c 1 alpha Stable Foliation including the classical lorenz attractor
arXiv: Dynamical Systems, 2015Co-Authors: Vitor Araujo, Ian MelbourneAbstract:We prove exponential decay of correlations for a class of $C^{1+\alpha}$ uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open set of geometric Lorenz attractors. As a special case, we show that the classical Lorenz attractor is robustly exponentially mixing.
Vitor Araujo - One of the best experts on this subject based on the ideXlab platform.
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mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth Stable Foliation
Advances in Mathematics, 2019Co-Authors: Vitor Araujo, Ian MelbourneAbstract:Abstract Over the last 10 years or so, advanced statistical properties, including exponential decay of correlations, have been established for certain classes of singular hyperbolic flows in three dimensions. The results apply in particular to the classical Lorenz attractor. However, many of the proofs rely heavily on the smoothness of the Stable Foliation for the flow. In this paper, we show that many statistical properties hold for singular hyperbolic flows with no smoothness assumption on the Stable Foliation. These properties include existence of SRB measures, central limit theorems and associated invariance principles, as well as results on mixing and rates of mixing. The properties hold equally for singular hyperbolic flows in higher dimensions provided the center-unStable subspaces are two-dimensional.
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existence and smoothness of the Stable Foliation for sectional hyperbolic attractors
Bulletin of The London Mathematical Society, 2017Co-Authors: Vitor Araujo, Ian MelbourneAbstract:We prove the existence of a contracting invariant topological Foliation in a full neighbourhood for partially hyperbolic attractors. Under certain bunching conditions, it can then be shown that this Stable Foliation is smooth. Specialising to sectional hyperbolic attractors, we give a verifiable condition for bunching. In particular, we show that the Stable Foliation for the classical Lorenz equation (and nearby vector fields) is better than C1, which is crucial for recent results on exponential decay of correlations. In fact, the Foliation is at least C1.278
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exponential decay of correlations for nonuniformly hyperbolic flows with a c 1 alpha c 1 α Stable Foliation including the classical lorenz attractor
Annales Henri Poincaré, 2016Co-Authors: Vitor Araujo, Ian MelbourneAbstract:We prove exponential decay of correlations for a class of \({C^{1+\alpha}}\) uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open set of geometric Lorenz attractors. As a special case, we show that the classical Lorenz attractor is robustly exponentially mixing.
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exponential decay of correlations for nonuniformly hyperbolic flows with a c 1 alpha Stable Foliation
arXiv: Dynamical Systems, 2015Co-Authors: Vitor Araujo, Ian MelbourneAbstract:We prove exponential decay of correlations for a class of C^{1+\alpha} uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open and dense set of geometric Lorenz attractors in a neighborhood of the classical Lorenz attractor.
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exponential decay of correlations for nonuniformly hyperbolic flows with a c 1 alpha Stable Foliation including the classical lorenz attractor
arXiv: Dynamical Systems, 2015Co-Authors: Vitor Araujo, Ian MelbourneAbstract:We prove exponential decay of correlations for a class of $C^{1+\alpha}$ uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open set of geometric Lorenz attractors. As a special case, we show that the classical Lorenz attractor is robustly exponentially mixing.
Ko Honda - One of the best experts on this subject based on the ideXlab platform.
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TOPOLOGICAL ENTROPY FOR REEB VECTOR FIELDS IN DIMENSION THREE VIA OPEN BOOK DECOMPOSITIONS
Ecole Polytechnique, 2019Co-Authors: Alves Marcelo, Colin Vincent, Ko HondaAbstract:International audienceGiven an open book decomposition of a contact three man-ifold (M, ξ) with pseudo-Anosov monodromy and fractional Dehn twist coefficient c = k n , we construct a Legendrian knot Λ close to the Stable Foliation of a page, together with a small Legendrian pushoff Λ. When k ≥ 5, we apply the techniques of [CH2] to show that the strip Legen-drian contact homology of Λ → Λ is well-defined and has an exponential growth property. The work [Al2] then implies that all Reeb vector fields for ξ have positive topological entropy
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topological entropy for reeb vector fields in dimension three via open book decompositions
Journal de l'Ecole Polytechnique - Mathematiques, 2019Co-Authors: Marcelo R R Alves, Vincent Colin, Ko HondaAbstract:Given an open book decomposition of a contact three man-ifold (M, ξ) with pseudo-Anosov monodromy and fractional Dehn twist coefficient c = k n , we construct a Legendrian knot Λ close to the Stable Foliation of a page, together with a small Legendrian pushoff Λ. When k ≥ 5, we apply the techniques of [CH2] to show that the strip Legen-drian contact homology of Λ → Λ is well-defined and has an exponential growth property. The work [Al2] then implies that all Reeb vector fields for ξ have positive topological entropy.
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Topological entropy for Reeb vector fields in dimension three via open book decompositions
2017Co-Authors: Alves Marcelo, Colin Vincent, Ko HondaAbstract:Given an open book decomposition of a contact three man-ifold (M, $\xi$) with pseudo-Anosov monodromy and fractional Dehn twist coefficient c = k n, we construct a Legendrian knot $\Lambda$ close to the Stable Foliation of a page, together with a small Legendrian pushoff $\Lambda$. When k $\ge$ 5, we apply the techniques of [CH2] to show that the strip Legen-drian contact homology of $\Lambda$ $\rightarrow$ $\Lambda$ is well-defined and has an exponential growth property. The work [Al2] then implies that all Reeb vector fields for $\xi$ have positive topological entropy
Marks Ruziboev - One of the best experts on this subject based on the ideXlab platform.
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on the statistical stability of lorenz attractors with a c 1 alpha Stable Foliation
arXiv: Dynamical Systems, 2018Co-Authors: Wael Bahsoun, Marks RuziboevAbstract:We prove statistical stability for a family of Lorenz attractors with a C1+α Stable Foliation.
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on the stability of statistical properties for lorenz attractors with a c 1 alpha Stable Foliation
arXiv: Dynamical Systems, 2017Co-Authors: Wael Bahsoun, Marks RuziboevAbstract:We prove statistical stability and robustness of the variance in the Central Limit Theorem (CLT) for a family of Lorenz attractors with a $C^{1+\alpha}$ Stable Foliation.
Wael Bahsoun - One of the best experts on this subject based on the ideXlab platform.
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on the statistical stability of lorenz attractors with a c 1 alpha Stable Foliation
arXiv: Dynamical Systems, 2018Co-Authors: Wael Bahsoun, Marks RuziboevAbstract:We prove statistical stability for a family of Lorenz attractors with a C1+α Stable Foliation.
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on the stability of statistical properties for lorenz attractors with a c 1 alpha Stable Foliation
arXiv: Dynamical Systems, 2017Co-Authors: Wael Bahsoun, Marks RuziboevAbstract:We prove statistical stability and robustness of the variance in the Central Limit Theorem (CLT) for a family of Lorenz attractors with a $C^{1+\alpha}$ Stable Foliation.