On solenoidal distribution of infinite ω-limit sets

Abstract

The aim of this paper is to present a structure result on the infinite ω-limit sets of continuous triangular mappings defined on the unit square, i.e. maps of the form (x, y)→(f(x), g(x, y)) having zero topological entropy in the base map f and on the fibres defined over the periodic points of f. We show for this class of systems that the infinite ω-limit sets of the points of the form (x, y), where x is a periodic point of f, have a solenoidal distribution. It extends the results by Smítal [J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), pp. 269–282] on zero topological entropy of continuous interval maps. An application is presented.

Keywords:

• ω-limit set
• triangular mapping
• periodic points
• solenoidal structure

2000 AMS Subject Classification :

• 37B20
• 37B99

Acknowledgements

The first author was supported in part by MCYT grant number MTM2005-03860 and Fundación Séneca, grant number 00684-FI-04. The second author was supported in part by MCYT grant numbers MTM2005-03860 and MTM2005-06098-C02-01; and Fundación Séneca, grant number 00684-FI-04.