Stable Minimality of Expanding Foliations
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Stable Minimality of Expanding Foliations Gabriel Núñez1,2 · Jana Rodriguez Hertz3,4
Received: 29 May 2020 / Revised: 6 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We prove that generically in Diff 1m (M), if an expanding f -invariant foliation W of dimension u is minimal and there is a periodic point of unstable index u, the foliation is stably minimal. By this we mean there is a C 1 -neighborhood U of f such that for all C 2 -diffeomorphisms g ∈ U , the g-invariant continuation of W is minimal. In particular, all such g are topologically mixing. Moreover, all such g have a hyperbolic ergodic component of the volume measure m which is essentially dense. This component is, in fact, Bernoulli. We provide new examples of diffeomorphisms with stably minimal expanding foliations which are not partially hyperbolic. Keywords Minimal foliation · Stable minimality · Stable ergodicity
1 Introduction In this paper, we look for mechanisms activating the stable minimality of an expanding invariant foliation. From now on, let M be a closed Riemannian manifold, and let f be a C 1 -diffeomorphisms in M preserving a smooth volume m. An f -invariant foliation is expanding if it is tangent to a D f -invariant sub-bundle E of the tangent bundle T M such
GN was supported by Agencia Nacional de Investigación e Innovación. The research that gives rise to the results presented in this publication received funds from the Agencia Nacional de Investigación e Innovación under the code POS_NAC_2014_1_102348 JRH was supported by NSFC 11871262 and NSFC 11871394.
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Jana Rodriguez Hertz [email protected] Gabriel Núñez [email protected]
1
IMERL, Facultad de Ingeniería, Universidad de la República, Julio Herrera y Reissig 565, 11300 Montevideo, Uruguay
2
Departamento de matemática, Facultad de Ingeniería y Tecnologías, Universidad Católica del Uruguay, Comandante Braga 2715, 11600 Montevideo, Uruguay
3
Department of Mathematics, Southern University of Science and Technology of China, No 1088, xueyuan Rd., Xili, Nanshan District, Shenzhen 518055, Guangdong, China
4
SUSTech International Center for Mathematics, Shenzhen, China
123
Journal of Dynamics and Differential Equations
that D f (x)v > 1 for all unit vectors v ∈ E x , for every x ∈ M. A foliation is minimal if every leaf of the foliation is dense. An f -invariant foliation W is stably minimal if there exists a C 1 -neighborhood U ( f ) of f in Diff 1m (M) such that (1) For each g ∈ U there exists a g-invariant foliation Wg such that the fiber bundle T Wg varies continuously for g ∈ U ( f ), where W f = W (2) Wg is minimal for all g ∈ U ( f ) ∩ Diff 2m (M) With this definition, a stably minimal f -invariant foliation could be not minimal. However, if f ∈ Diff 2m (M), every stably minimal f -invariant foliation is minimal. Note that minimality of an invariant foliation is a G δ -property under condition (1) above; hence, the generic stably minimal f -invariant foliation will be minimal, even if f is only C 1 . We obtain the f
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