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Strange triangular maps of the square

Published online by Cambridge University Press:  17 April 2009

G.L. Forti ,
L. Paganoni  and
J. Smítal
G.L. Forti
Affiliation:
Dipartimento de Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133 Milano, Italia
L. Paganoni
Affiliation:
Dipartimento de Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133 Milano, Italia
J. Smítal
Affiliation:
Institute of Mathematics, Comenius University, 84215 Bratislava, Slovakia and Institute of Mathematics, Silesian University, 74061 Opava, Czech Republic
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Abstract

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We show that continuous triangular maps of the square I1, F: (x, y) → (f(x), g(x, y)), exhibit phenomena impossible in the one-dimensional case. In particular: (1) A triangular map F with zero topological entropy can have a minimal set containing an interval {a} × I, and can have recurrent points that are not uniformly recurrent; this solves two problems by S.F. Kolyada.

(2) In the class of mappings satisfying Per(F) = Fix(F), there are non-chaotic maps with positive sequence topological entropy and chaotic maps with zero sequence topological entropy.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 51 , Issue 3 , June 1995 , pp. 395 - 415
Copyright
Copyright © Australian Mathematical Society 1995

References

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