Nonlinear Dynamics

, Volume 84, Issue 1, pp 163–169

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

• Antonio Linero Bas
• Gabriel Soler López
Original Paper

Abstract

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 $$\varphi _j=f_{\sigma (j)}\circ \ldots \circ f_{\sigma ^n(j)},\, j\in \{1,\ldots ,n\}.$$

Keywords

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

Notes

Acknowledgments

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).

References

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)
3. 3.
Balibrea, F., Linero, A.: Periodic structure of $$\sigma$$-permutation maps on $$I^{n}$$. Aequ. Math. 62, 265–279 (2001)
4. 4.
Banks, J.: Regular periodic decompositions for topologically transitive maps. Ergod. Theory Dyn. Syst. 17, 505–529 (1997)
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)
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)
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)
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)