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

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  • 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 $I_0$.
    Mathematics Subject Classification: 37B20, 37B99.


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