Geometric expansion, Lyapunov exponents and foliations

Saghin, Radu; Xia, Zhihong

doi:10.1016/j.anihpc.2008.07.001

Cited by 34 publications

(34 citation statements)

References 13 publications

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“…The topological invariant was introduced in Saghin and Xia [5], where one can find more details and some other applications of the invariant. Currents, topological and geometric growth In this section, we will define some topological invariants for diffeomorphisms with uniformly expanding (or contracting) foliations.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Topological entropy and partially hyperbolic diffeomorphisms

Hua

Saghin

Xia

2008

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385707000405 How to cite this article: YONGXIA HUA, RADU SAGHIN and ZHIHONG XIA (2008). Topological entropy and partially hyperbolic diffeomorphisms.Abstract. We consider partially hyperbolic diffeomorphisms on compact manifolds. We define the notion of the unstable and stable foliations stably carrying some unique nontrivial homologies. Under this topological assumption, we prove the following two results: if the center foliation is one-dimensional, then the topological entropy is locally a constant; and if the center foliation is two-dimensional, then the topological entropy is continuous on the set of all C ∞ diffeomorphisms. The proof uses a topological invariant we introduced, Yomdin's theorem on upper semi-continuity, Katok's theorem on lower semi-continuity for two-dimensional systems, and a refined Pesin-Ruelle inequality we proved for partially hyperbolic diffeomorphisms.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Topological entropy and partially hyperbolic diffeomorphisms

Hua

Saghin

Xia

2008

Ergod. Th. Dynam. Sys.

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“…In [22], the authors need unique homological data for the strong unstable foliation and they prove that it is the case when f is closed to its linearization.…”

Section: Theorem 24 Let F Be a Cmentioning

confidence: 99%

Regularity of foliations and Lyapunov exponents of partially hyperbolic dynamics on 3-torus

Micena

Tahzibi²

2013

Nonlinearity

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“…In the setting of DA diffeomorphisms there have been some results concerning the nonabsolute continuity of the center foliation (for instance, [7,14]), but no further description. Theorem 1.1 gives more information than merely non-absolute continuity.…”

Section: R Varãomentioning

confidence: 98%

Center foliation: absolute continuity, disintegration and rigidity

Varão¹

2014

Ergod. Th. Dynam. Sys.

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In this paper we address the issues of absolute continuity for the center foliation (as well as the disintegration on the non-absolute continuous case) and rigidity of volume preserving partially hyperbolic diffeomorphisms isotopic to a linear Anosov on $\mathbb T^3$. It is shown that the disintegration of volume on center leaves may be neither atomic nor Lebesgue. It is also obtained results concerning the atomic disintegration. Moreover, the absolute continuity of the center foliation does not imply smooth conjugacy with its linearization. Imposing stronger conditions besides absolute continuity on the center foliation, smooth conjugacy is obtained

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Geometric expansion, Lyapunov exponents and foliations

Cited by 34 publications

References 13 publications

Topological entropy and partially hyperbolic diffeomorphisms

Topological entropy and partially hyperbolic diffeomorphisms

Regularity of foliations and Lyapunov exponents of partially hyperbolic dynamics on 3-torus

Center foliation: absolute continuity, disintegration and rigidity

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