Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center
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. ARTICLES .
September 2020 Vol. 63 No. 9: 1647–1670 https://doi.org/10.1007/s11425-019-1751-2
Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center To the Memory of Professor Shantao Liao
Christian Bonatti1 & Jinhua Zhang2,∗ 1Institut
de Math´ ematiques de Bourgogne, Universit´ e de Bourgogne, Dijon 21104, France; of Mathematical Sciences, Beihang University, Beijing 100191, China
2School
Email: [email protected], [email protected] Received May 5, 2019; accepted July 21, 2020; published online August 21, 2020
Abstract
In this paper, we study transitive partially hyperbolic diffeomorphisms with one-dimensional topo-
logically neutral center, meaning that the length of the iterate of small center segments remains small. Such systems are dynamically coherent. We show that there exists a continuous metric along the center foliation which is invariant under the dynamics. As an application, we classify the transitive partially hyperbolic diffeomorphisms on 3-manifolds with topologically neutral center. Keywords MSC(2010)
partial hyperbolicity, dynamical coherence, conjugacy, transitivity, neutral 37D30, 37C15, 37E05, 57M60
Citation: Bonatti C, Zhang J H. Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center. Sci China Math, 2020, 63: 1647–1670, https://doi.org/10.1007/s11425-019-1751-2
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Introduction
A C 1 diffeomorphism f on a closed manifold M is partially hyperbolic if there exists a Df -invariant splitting T M = Es ⊕ Ec ⊕ Eu such that E s is uniformly contracting, E u is uniformly expanding and E c has the intermediate behavior; to be precise, there exists an integer N ∈ N such that for any x ∈ M , we have the following: • Contraction and expansion. ∥Df N |E s (x) ∥
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