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Stable Minimality of Expanding Foliations

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Abstract

We prove that generically in \({\text {Diff}}^{1}_{m}(M)\), 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 \({\mathcal {U}}\) of f such that for all \(C^{2}\)-diffeomorphisms \(g\in {\mathcal {U}}\), 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.

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Acknowledgements

We want to thank R. Ures for useful suggestions and discussions that helped improve this paper.

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Authors and Affiliations

  1. IMERL, Facultad de Ingeniería, Universidad de la República, Julio Herrera y Reissig 565, 11300, Montevideo, Uruguay

    Gabriel Núñez

  2. Departamento de matemática, Facultad de Ingeniería y Tecnologías, Universidad Católica del Uruguay, Comandante Braga 2715, 11600, Montevideo, Uruguay

    Gabriel Núñez

  3. Department of Mathematics, Southern University of Science and Technology of China, No 1088, xueyuan Rd., Xili, Nanshan District, Shenzhen, 518055, Guangdong, China

    Jana Rodriguez Hertz

  4. SUSTech International Center for Mathematics, Shenzhen, China

    Jana Rodriguez Hertz

Authors
  1. Gabriel Núñez
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  2. Jana Rodriguez Hertz
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Correspondence to Jana Rodriguez Hertz.

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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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Núñez, G., Rodriguez Hertz, J. Stable Minimality of Expanding Foliations. J Dyn Diff Equat 33, 2075–2089 (2021). https://doi.org/10.1007/s10884-020-09884-x

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