On skew-product maps with the base having a closed set of periodic points

Abstract

Sharkovskii˘ and Kolyada reported the problem of the characterization of skew-product maps having zero topological entropy. It is known that, even under additional assumptions, this aim has not been attained. Guirao and Chudziak partially solved this problem for the class of skew-product maps with the base map having a closed set of periodic points. The present paper has two aims for this class of map. On the one hand, to improve that solution showing the equivalence between the property ‘of having zero topological entropy’ and the fact of ‘not being Li–Yorke chaotic in the union of the ω-limit sets of recurrent points’. On the other hand, we show that the properties ‘of having a closed set of periodic points’ and ‘all non-wandering points are periodic’ are not mutually equivalent properties. In doing this we disprove a result of Efremova.

Keywords:

• Dynamical system
• Topological dynamics
• Triangular maps
• Morse–Smale maps; Omega-limit sets

Acknowledgements

The authors are very grateful to the referees, in particular to referee 2, for the valuable suggestions presented in a constructive spirit. This study was partially supported by MCYT grants MTM2005-03860 and MTM2005-06098-C02-01, by Fundación Séneca, grant 00684-FI-04, and PAI06-0114 and PBC05-011-3 (JCCM)