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**Robust density of periodic orbits for skew products with high dimensional fiber.**
*(English)*
Zbl 1343.37013

Summary: We consider step and soft skew products over the Bernoulli shift which have an \(m\)-dimensional closed manifold as a fiber. It is assumed that the fiber maps Hölder continuously depend on a point in the base. We prove that, in the space of skew product maps with this property, there exists an open domain such that maps from this open domain have dense sets of periodic points that are attracting and repelling along the fiber. Moreover, robust properties of invariant sets of diffeomorphisms, including the coexistence of dense sets of periodic points with different indices, are obtained.

### MSC:

37C05 | Dynamical systems involving smooth mappings and diffeomorphisms |

37A25 | Ergodicity, mixing, rates of mixing |

37C27 | Periodic orbits of vector fields and flows |

37D30 | Partially hyperbolic systems and dominated splittings |

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\textit{F. H. Ghane} et al., Abstr. Appl. Anal. 2013, Article ID 539736, 7 p. (2013; Zbl 1343.37013)

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### References:

[1] | Gorodetski, A.; Ilyashenko, Y. S., Some properties of skew products over a horseshoe and solenoid, Proceedings of the Steklov Institute of Mathematics, 231, 96-118 (2000) · Zbl 1002.37017 |

[2] | Gorodetski, A. S.; Ilyashenko, Y. S., Some new robust properties of invariant sets and attractors of dynamical systems, Funktsional’nyi Analiz i Ego Prilozheniya, 33, 2, 16-30 (1999) · Zbl 0939.37015 · doi:10.1007/BF02465190 |

[3] | Homburg, A. J.; Nassiri, M., Robust minimality of iterated function systems with two generators, Ergodic Theory and Dynamical Systems, 1-6 (2013) |

[4] | Ghane, F. H.; Nazari, M.; Saleh, M.; Shabani, Z., Attractors and their invisible parts for skew products with high dimensional fiber, International Journal of Bifurcation and Chaos, 22, 8 (2012) · Zbl 1258.37022 · doi:10.1142/S0218127412501829 |

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