Exceptional minimal sets of <i>C</i><sup>1+α</sup>-group actions on the circle

Hurder, Steven

doi:10.1017/s0143385700006271

Cited by 10 publications

(17 citation statements)

References 18 publications

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“…We point out one application of Proposition 5.8, which complements the main result of [42]. THEOREM 1.3.…”

Section: Introductionsupporting

confidence: 57%

“…The plaque chain P determines a closed loop γ P based at x in the leaf L x , and the transverse holonomy along γ P is linearly expanding in some open neighborhood of x. Such transversally hyperbolic elements of the leaf holonomy have a fundamental role in the study of foliation dynamics, in particular in the works by Sacksteder [70], by Bonatti, Langevin and Moussu [4], and the works [40,42]. However, given x ∈ E + (F) there may not be a closed leafwise loop with infinitesimally expansive holonomy at x.…”

Section: 2mentioning

confidence: 99%

“…We note that the existence of a hyperbolic contraction can also be deduced using the foliation geodesic flow methods introduced in [42], though that method does not yield estimates on the size of the domain of the hyperbolic element in the foliation pseudogroup.…”

Section: Uniform Hyperbolic Expansionmentioning

confidence: 99%

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Dynamics and the Godbillon–Vey class of $C^1$ foliations

Hurder¹,

Langevin²

2018

J. Math. Soc. Japan

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Let F be a codimension-one, C 2 -foliation on a manifold M without boundary. In this work we show that if the Godbillon-Vey class GV (F ) ∈ H 3 (M ) is non-zero, then F has a hyperbolic resilient leaf. Our approach is based on methods of C 1 -dynamical systems, and does not use the classification theory of C 2 -foliations. We first prove that for a codimension-one C 1foliation with non-trivial Godbillon measure, the set of infinitesimally expanding points E(F ) has positive Lebesgue measure. We then prove that if E(F ) has positive measure for a C 1 -foliation, then F must have a hyperbolic resilient leaf, and hence its geometric entropy must be positive. The proof of this uses a pseudogroup version of the Pliss Lemma. The first statement then follows, as a C 2 -foliation with non-zero Godbillon-Vey class has non-trivial Godbillon measure. These results apply for both the case when M is compact, and when M is an open manifold.

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“…We point out one application of Proposition 5.8, which complements the main result of [42]. THEOREM 1.3.…”

Section: Introductionsupporting

confidence: 57%

Section: 2mentioning

confidence: 99%

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Dynamics and the Godbillon–Vey class of $C^1$ foliations

Hurder¹,

Langevin²

2018

J. Math. Soc. Japan

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“…Some closely related versions were obtained by Hurder in [119,120,122,123] via dynamical methods coming from the so-called foliated Pesin's theory. The optimal and definitive versions (obtained by the author in collaboration with Deroin and Kleptsyn in [65]) make a strong use of probabilistic arguments.…”

Section: The C 1 Version For Pseudo-groupsmentioning

confidence: 98%

Groups of Circle Diffeomorphisms

Navas¹

2011

150

171

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The original version (in Spanish) of this text was prepared for a mini-course in Antofagasta, Chile. Subsequently, enlarged and revised versions were published in the series Monografías del IMCA (Perú) and Ensaios Matemáticos (Brasil). This translation arose from the necessity of making this text accessible to a larger audience. I would like to thank Juan Rivera-Letelier for his invitation to the II Workshop on Dynamical Systems ( 2001), for which the original version was prepared, and Roger Metzger for his invitation to IMCA (2006), where part of this material was presented. I would also like to thank both Étienne Ghys and Maria Eulália Vares for motivating me and allowing me to publish the revised Spanish version in Brasil, as well as all the people who strongly encouraged me to conclude this English version.This work was partially supported by CONICYT and PBCT via the Research Network on Low Dimensional Dynamics. I would also to acknowledge the support of both the UMPA Department of the École Normale Supérieure de Lyon, where the idea of writing this text was born during my PhD thesis, and the Institut des Hautes Études Scientifiques, where the original notes started taking its definite form while I benefited from a one-year postdoctoral position.This work owes a lot to many of my colleges. Several remarks spread throughout the text and the content of some examples and exercises were born during fruitful and quite stimulating discussions. It is then a pleasure to thank Sylvain Crovisier (Proposition 4.2.25),

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“…The presence of leaves with exponential growth rates for foliations can only occur when the leaves have dimension at least two. It was observed already in various works in the 1960's that the existence of leaves of exponential growth for foliations make the study of foliation dynamics "exceptional" [159,262,263,273,275,276].…”

Section: Orbit Growth and The Følner Conditionmentioning

confidence: 99%

Classifying foliations

Hurder¹

2009

Foliations, Geometry, and Topology

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Abstract. We give a survey of the approaches to classifying foliations, starting with the Haefliger classifying spaces and the various results and examples about the secondary classes of foliations. Various dynamical properties of foliations are introduced and discussed, including expansion rate, local entropy, and orbit growth rates. This leads to a decomposition of the foliated space into Borel or measurable components with these various dynamical types. The dynamical structure is compared with the classification via secondary classes.1991 Mathematics Subject Classification. Primary 22F05, 37C85, 57R20, 57R32, 58H05, 58H10; Secondary 37A35, 37A55, 37C35 .

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Exceptional minimal sets of C^1+α-group actions on the circle

Cited by 10 publications

References 18 publications

Dynamics and the Godbillon–Vey class of $C^1$ foliations

Dynamics and the Godbillon–Vey class of $C^1$ foliations

Groups of Circle Diffeomorphisms

Classifying foliations

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Exceptional minimal sets of C1+α-group actions on the circle

Cited by 10 publications

References 18 publications

Dynamics and the Godbillon–Vey class of $C^1$ foliations

Dynamics and the Godbillon–Vey class of $C^1$ foliations

Groups of Circle Diffeomorphisms

Classifying foliations

Contact Info

Product

Resources

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Exceptional minimal sets of C^1+α-group actions on the circle