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2002, 8(4): 983-994. doi: 10.3934/dcds.2002.8.983

A triangular map on $I^{2}$ whose $\omega$-limit sets are all compact intervals of $\{0\}\times I$

1. 

Departamento de Matemáticas, Universidad de Murcia, 30100-Murcia, Spain

2. 

Departamento de Fundamentos del Análisis Económico, Universidad de Alicante, 03071-Alicante, Spain

3. 

Departamento de Matemáticas, I.E.S. J. Ibáñez Martín, 30800-Lorca(Murcia), Spain

Received  July 2001 Revised  March 2002 Published  July 2002

In this paper we construct a triangular map $F$ on $I^2$ which holds the following property. For each $[a,b]\subseteq I=[0,1]$, $a\leq b$, there exists $(p,q)\in I^2$ \ $I_0$ such that $\omega_F(p,q)=$ {0} $\times [a,b]\subset I_0$ where $I_0=${0}$\times I$. Moreover, for each $(p,q)\in I^{2}$, the set $\omega_F(p,q)$ is exactly {0} $\times J$ where $J\subset I$ is a compact interval degenerate or not. So, we describe completely the family $\mathcal W(F)=${$\omega_F(p,q):(p,q)\in I^2$} and establish $\mathcal W(F)$ as the set of all compact interval, degenerate or not, of $I_0$.
Citation: Francisco Balibrea, J.L. García Guirao, J.I. Muñoz Casado. A triangular map on $I^{2}$ whose $\omega$-limit sets are all compact intervals of $\{0\}\times I$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 983-994. doi: 10.3934/dcds.2002.8.983
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