Nonlinear Dynamics

, Volume 84, Issue 1, pp 163–169 | Cite as

A splitting result on transitivity for a class of n-dimensional maps

Original Paper


We obtain some results on transitivity for cyclically permuted direct product maps, that is, maps of the form \(F\left( x_{1},x_{2},\ldots ,x_{n}\right) =\left( f_{\sigma (1)}\left( x_{\sigma (1)}\right) ,f_{\sigma (2)}\left( x_{\sigma (2)}\right) ,\ldots ,f_{\sigma (n)}\left( x_{\sigma (n)}\right) \right) \), defined from the Cartesian product \(I^n\) onto itself, where \(I=[0,1]\), \(\sigma \) is a cyclic permutation of \(\{1,2,\ldots ,n\}\) \((n\ge 2)\) and each map \(f_{\sigma (j)}:I\rightarrow I\) is continuous, \(j\in \{1,\ldots ,n\}\). In particular, we prove that for \(n\ge 3\) the transitivity of F is equivalent to the total transitivity, and if \(n=2\), we give a splitting result for transitive maps. Moreover, we extend well-known properties of transitivity from interval maps to cyclically permuted direct product maps. To do it, we use the strong link between F and the compositions \(\varphi _j=f_{\sigma (j)}\circ \ldots \circ f_{\sigma ^n(j)},\, j\in \{1,\ldots ,n\}.\)


Cyclic permutation Cyclically permuted direct product map Transitivity Mixing Weakly mixing Total transitivity Splitting 



The authors thank the referees for their interesting suggestions which have improved this final version. This paper has been partially supported by grant 19294/PI/14 (Fundación Séneca, Comunidad Autónoma de la Región de Murcia, Spain) and by grants MTM2011-23221 and MTM2014-52920-P (Ministerio de Economía y Competitividad, Spain).


  1. 1.
    Agronsky, S., Ceder, J.: What sets can be \(\omega \) limit sets in \(E^{n}\)? Real Anal. Exch. 17, 97–109 (1991–1992)Google Scholar
  2. 2.
    Alsedà, L.I., del Río, M.A., Rodríguez, J.A.: A splitting theorem for transitive maps. J. Math. Anal. Appl. 232, 359–375 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Balibrea, F., Linero, A.: Periodic structure of \(\sigma \)-permutation maps on \(I^{n}\). Aequ. Math. 62, 265–279 (2001)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Banks, J.: Regular periodic decompositions for topologically transitive maps. Ergod. Theory Dyn. Syst. 17, 505–529 (1997)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Block, L., Coppel, W.A.: Dynamics in one dimension. Lecture Notes in Mathematics, vol. 1513. Springer, Berlin (1992)Google Scholar
  6. 6.
    Dana, R.A., Montrucchio, L.: Dynamical complexity in duopoly games. J. Econ. Theory 40, 40–56 (1986)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Franke, J.E., Yakubu, A.A.: Attenuant cycles in periodically forced discrete-time age-structured population models. J. Math. Anal. Appl. 316, 69–86 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kolyada, S., Snoha, L.: Some aspects of topological transitivity–a survey, Iteration Theory (ECIT 94) (Opava). Grazer Math. Ber. 334, 3–35 (1997)MathSciNetMATHGoogle Scholar
  9. 9.
    Linero Bas, A., Soler López, G.: A note on the dynamics of cyclically permuted direct product maps. Topol. Appl. (2015) (forthcoming)Google Scholar
  10. 10.
    Puu, T.: Nonlinear Economic Dynamics. Springer, Berlin (1997)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de MurciaMurciaSpain
  2. 2.Departamento de Matemática Aplicada y EstadísticaUniversidad Politécnica de CartagenaCartagenaSpain

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