Triangular maps with all periods and no infinite ω-limit set containing periodic points

Abstract

Our main result is an example of a triangular map of the unite square, F ( x , y ) = ( f ( x ) , g x ( y ) ) , possessing periodic orbits of all periods and such that no infinite ω-limit set of F contains a periodic point. We also show that there is a triangular map F of type 2 ∞ monotone on the fibres such that any recurrent point of F is uniformly recurrent and F restricted to the set of its recurrent points is chaotic in the sense of Li and Yorke. For a continuous map φ of the interval there is a long list of more than 50 conditions characterizing zero topological entropy, including, e.g., conditions (i) φ is of type 2 ∞ , (ii) every recurrent point of φ is uniformly recurrent, (iii) φ restricted to the set of chain recurrent points is not chaotic in the sense of Li and Yorke, (iv) no infinite ω-limit set contains a cycle. The problem presented by A.N. Sharkovsky in the eighties is to decide which of these conditions remain equivalent in the class of triangular maps. Our second example completes the results obtained, e.g., by Forti et al. (1999), Kočan (2003) and Šindelářová (2003), concerning triangular maps monotone on the fibres. The first example, with a more sophisticated proof, contributes to a more difficult problem of classification of general triangular maps, which is still not completely solved; the main partial results have been obtained by Kolyada (1992).