Differential properties and attracting sets of a~simplest skew product of interval maps

L. S. Efremova

Sbornik. Mathematics, Tome 201 (2010) no. 6, pp. 873-907

Résumé

For a skew product of interval maps with a closed set of periodic points, the dependence of the structure of its $\omega$-limit sets on its differential properties is investigated. An example of a map in this class is constructed which has the maximal differentiability properties (within a certain subclass) with respect to the variable $x$, is $C^1$-smooth in the $y$-variable and has one-dimensional $\omega$-limit sets. Theorems are proved that give necessary conditions for one-dimensional $\omega$-limit sets to exist. One of them is formulated in terms of the divergence of the series consisting of the values of a function of $x$; this function is the $C^{0}$-norm of the deviation of the restrictions of the fibre maps to some nondegenerate closed interval from the identity on the same interval. Another theorem is formulated in terms of the properties of the partial derivative with respect to $x$ of the fibre maps. A complete description is given of the $\omega$-limit sets of certain class of $C^1$-smooth skew products satisfying some natural conditions. Bibliography: 33 titles.

Détail

Keywords: skew product, periodic point, $\omega$-limit set
Mots-clés : divergent series.