Abstract and Applied Analysis

Robust Density of Periodic Orbits for Skew Products with High Dimensional Fiber

Fatemeh Helen Ghane, Mahboubeh Nazari, Mohsen Saleh, and Zahra Shabani

Full-text: Open access

Abstract

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.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 539736, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393512075

Digital Object Identifier
doi:10.1155/2013/539736

Mathematical Reviews number (MathSciNet)
MR3111805

Zentralblatt MATH identifier
1343.37013

Citation

Ghane, Fatemeh Helen; Nazari, Mahboubeh; Saleh, Mohsen; Shabani, Zahra. Robust Density of Periodic Orbits for Skew Products with High Dimensional Fiber. Abstr. Appl. Anal. 2013 (2013), Article ID 539736, 7 pages. doi:10.1155/2013/539736. https://projecteuclid.org/euclid.aaa/1393512075


Export citation

References

  • A. Gorodetski and Y. S. Ilyashenko, “Some properties of skew products over a horseshoe and solenoid,” Proceedings of the Steklov Institute of Mathematics, vol. 231, pp. 96–118, 2000.
  • A. S. Gorodetski and Y. S. Ilyashenko, “Some new robust properties of invariant sets and attractors of dynamical systems,” Funktsional'nyi Analiz i Ego Prilozheniya, vol. 33, no. 2, pp. 16–30, 1999.
  • A. J. Homburg and M. Nassiri, “Robust minimality of iterated function systems with two generators,” Ergodic Theory and Dynamical Systems, pp. 1–6, 2013.
  • F. H. Ghane, M. Nazari, M. Saleh, and Z. Shabani, “Attractors and their invisible parts for skew products with high dimensional fiber,” International Journal of Bifurcation and Chaos, vol. 22, no. 8, Article ID 1250182, 16 pages, 2012.