Abstract
We obtain sufficient conditions for total topological transitivity (transitivity of all iterations) for a class of C 3 skew products defined on cells in ℝn, n ≥ 2.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. S. Efremova and A. S. Fil’chenkov, “Topological transitivity of skew products on the plane with negative Schwarzian of the family of maps in the fibers,” Tr.MFTI 4 (4), 82–93 (2012).
L. S. Efremova and A. S. Fil’chenkov, “Boundary conditions for maps in fibers and topological transitivity of skew products of interval maps,” J. Math. Sci. (N. Y.) 208 (1), 109–114 (2015).
A. S. Fil’chenkov, Some Topologically Transitive Skew Products on Cells in Rn (n ≥ 2), Available from VINITI, No. 341-V2014 (2014) [in Russian].
W. de Melo and S. van Strien, One-Dimensional Dynamics, in Ergeb. Math. Grenzgeb. (3), Vol. 25 (Springer-Verlag, Berlin, 1996).
A. B. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge Univ. Press, Cambridge, 1995).
Ll. Alsedá, M. A. Del Río, and J. A. Rodríguez, “A survey on the relation between transitivity and dense periodicity for graph maps,” J. Difference Equ. Appl. 9 (3-4), 281–288 (2003).
A. Denjoy, “Sur les courbes defines par les équations différentielles à la surface du tore,” J. Math. Pures Appl. 11 (9), 333–375 (1932).
A. Denjoy, “Les trajectories à la surface du tore,” C. R. Acad. Sci. Paris 223, 5–8 (1946).
A. Denjoy, “Sur les caractéristiques à la surface du tore,” C. R. Acad. Sci. Paris 194, 830–833 (1932).
J. Guckenheimer, “Sensitive dependence to initial conditions for one-dimensional maps,” Comm. Math. Phys. 70 (2), 133–160 (1979).
D. Singer, “Stable orbits and bifurcations of maps of the interval,” SIAM J. Appl. Math. 35 (2), 260–267 (1978).
A. N. Sharkovskii, Yu. L. Maistrenko, and E. Yu. Romanenko, Difference Equations and Their Applications (Naukova Dumka, Kiev, 1986) [in Russian].
L. Block, J. Guckenheimer, M. Misiurewicz, and L. S. Young, “Periodic points and topological entropy of one-dimensional maps,” in Lecture Notes in Math. 819: Global Theory of Dynamical Systems (Springer-Verlag, Berlin, 1980), pp. 18–34.
O. M. Sharkovs’kii, “Fixed points and the center of a continuous mapping of the line into itself,” Dopovidi Akad. Nauk Ukrain. RSR 7, 865–868 (1964).
E. D’Aniello and T. H. Steele, “Approximating ω-limit sets with periodic orbits,” Aequationes Math. 75 (1-2), 93–102 (2008).
A. S. Fil’chenkov, “The skew product on n-dimensional cell with transitive but not totally transitive n-dimensional attractor,” Izv. Vyssh.Uchebn. Zaved. Mat., No. 6, 91–100 (2016) [Russian Math. (Iz. VUZ) 60 (6), 79–87 (2016)].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A. S. Fil’chenkov, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 1, pp. 109–124.
Rights and permissions
About this article
Cite this article
Fil’chenkov, A.S. On a class of totally topologically transitive skew products defined on cells in ℝn, n ≥ 2. Math Notes 102, 92–104 (2017). https://doi.org/10.1134/S0001434617070100
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434617070100