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

Francisco Balibrea; J.L. García Guirao; J.I. Muñoz Casado

doi:10.3934/dcds.2002.8.983

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2002, Volume 8, Issue 4: 983-994. Doi: 10.3934/dcds.2002.8.983

This issue Previous Article Explicit quasiconvexification for some cost functionals depending on derivatives of the state in optimal designing Next Article Higher order Melnikov function for a quartic hamiltonian with cuspidal loop

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
2.
Departamento de Fundamentos del Análisis Económico, Universidad de Alicante, 03071-Alicante
3.
Departamento de Matemáticas, I.E.S. J. Ibáñez Martín, 30800-Lorca(Murcia)

Received: July 2001

Revised: March 2002

Published: October 2002

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Abstract

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 $I_0$.

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Mathematics Subject Classification: 37B20, 37B99.

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