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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

L'ubomír Snoha 1, and  Vladimír Špitalský 1,

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.
Keywords: uniformly recurrent point, nonhomogeneous space, triangular map, chain recurrent point, Minimal system, topological entropy., minimal set, recurrent point, skew product.
Mathematics Subject Classification: Primary 37B05, 37B20, 37B40; Secondary 54H2.
Citation: L'ubomír Snoha, Vladimír Špitalský. Recurrence equals uniform recurrence does not imply zero entropy for triangular maps of the square. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 821-835. doi: 10.3934/dcds.2006.14.821
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