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A decomposition theorem for the space of \(C^1\)-smooth skew products with complicated dynamics of the quotient map. (English. Russian original) Zbl 1342.37040

Sb. Math. 204, No. 11, 1598-1623 (2013); translation from Mat. Sb. 204, No. 11, 55-82 (2013).
This paper is a natural extention of the author’s previous papers [J. Math. Sci., New York 105, No. 1, 1779–1798 (2001; Zbl 1193.37058); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat Obz. 67, 129–160 (1999); Theor. Math. Phys. 164, No. 3, 1208–1214 (2010; Zbl 1253.37043); translation from Teor. Mat. Fiz. 164, No. 3, 447–454 (2010)] on skew products of interval maps. The main result of the paper reviewed here is to decompose the space of \(C^1\)-smooth skew products into four nonempty pairwise disjoint subspaces when the quotient map (first component of the skew product) has complicated dynamics and is \(\Omega\)-stable in the \(C^1\)-norm (stability of the nonwandering set).

MSC:

37E05 Dynamical systems involving maps of the interval
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
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