ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXII, 2 (2003)
p. 245 – 251

Triangular Maps with the Chain Recurent Points Periodic
J. Kupka

Abstract.  Forti and Paganoni [Grazer Math. Ber. {\bf 339} (1999), 125--140] found a triangular map $F(x,y)=(f(x),g_x (y))$ from $I\times I$ into itself for which closed set of~periodic points is a proper subset of the set of chain recurrent points. We asked whether there is a characterization of triangular maps for which every chain recurrent point is periodic. We answer this question in positive by showing that, for a triangular map with closed set of periodic points and any posi\-tive real~$\varepsilon$, every $\varepsilon$-chain from a chain recurrent point to itself may be represented as a finite union of $\varepsilon$-chains whose all points either are periodic or form a nontrivial $\varepsilon$-chain of some one-dimensional map~$g_x$.

AMS subject classification:  37E99, 37B20, 26A18, 54H20;
Keywords:  Triangular maps, periodic points, chain recurrent points