October  2006, 14(4): 821-835. doi: 10.3934/dcds.2006.14.821

Recurrence equals uniform recurrence does not imply zero entropy for triangular maps of the square

1. 

Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovak Republic, Slovak Republic

Received  December 2004 Revised  September 2005 Published  January 2006

In 1992, S. Kolyada asked the question whether for triangular maps of the square zero topological entropy is equivalent to the fact that every recurrent point is uniformly recurrent. One of the implications was answered negatively by G.-L. Forti, L. Paganoni and J. Smítal in 1995. They showed that a zero entropy triangular map may have a recurrent point which is not uniformly recurrent. In this paper we show that neither the converse implication is true by constructing a triangular map of the square with positive topological entropy and with every chain recurrent point uniformly recurrent. In fact we first construct an appropriate minimal positive entropy system whose phase space is a nonhomogeneous subset of the square and then we extend it to a triangular map with required properties in the square.
Citation: L'ubomír Snoha, Vladimír Špitalský. Recurrence equals uniform recurrence does not imply zero entropy for triangular maps of the square. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 821-835. doi: 10.3934/dcds.2006.14.821
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