On omega-limit sets of antitriangular maps
by
Antonio Linero Bas
Universidad de Murcia. Departamento de Matemáticas
Coauthors: Francisco Balibrea Gallego (Universidad de Murcia), Jose S. Cánovas (Universidad Politécnica de Cartagena)

1  Introduction

Let C(X, X) be the set of continuous maps from the compact metric space X into itself. Let j Î C(X, X). The orbit of x Î X through j is the set {jⁿ(x)}_n=0^¥, where j⁰=Identity and jⁿ denotes the nth iterate of j, jⁿ=j°j^n-1, n Î N. For x Î X, the set w_j(x)={y Î X: ${n_i} _i Ì N such that j^n_i(x)\overset n_i® ¥® y} is called the omega-limit set of x by j. Moreover, Int(Y) denotes the interior of a subset Y Ì X.

In this talk we present some results concerning the topological structure of w-limit sets of a particular class of C(I², I²), I=[0, 1], the antitriangular maps

F(x, y)=(f2(y), f1(x)).

This type of two-dimensional maps appears closely related with an economical process so called Cournot duopoly. There is an abundant literature dealing with this model ([], [], ...).

2  Topological structure of w-limit sets

The topological structure of an w-limit set L of f Î C(I, I) is well known (consult [, Chapter V]). L is either a nowhere dense set (the closure of its interior is empty) or L=È_i=1ⁿJ_i, where J_i are nondegenerate closed subintervals of I such that f(J_i)=J_i+1(modn) and J_iÇJ_k=Æ if i ¹ k. Conversely, any set L of the above forms can be realized as an w-limit set for a suitable continuous interval map (see []).

The description of w-limit sets for f Î C(Iⁿ, Iⁿ), with n ³ 2, is an open problem, and only partial results are known ([], [], ...). It seems to be more reasonable to focus our attention on special classes of two-dimensional continuous maps. This was the strategy followed in [] in the case of triangular maps T(x, y)=(f(x), g(x, y)). Now, we are interested in studying the topological structure of w-limit sets for antitriangular maps on I². We obtain the following result.

Let F Î C(I², I²) be an antitriangular map. Let L be an w-limit set of F. Then either L is a nowhere dense set or L=È_i=1^pR_i, where R_i=[a_i1, a_i2]×[b_i1, b_i2] are periodic rectangles of F with Int(R_i)ÇInt(R_k)=Æ if j ¹ k.

It is remarkable to point out that we are able to describe w-limit sets with non-empty interior: The periodic rectangles {R_i}_i=1ⁿ produce an w-limit set of an antitriangular map if and only if they present an special distribution in the square completely equivalent to the description of finite periodic orbits given in []. This result generalizes empirical observations done in [], where periodic attracting rectangles are numerically showed.

Finally, we can also remark that in the case of w-limits of antitriangular maps with empty interior we obtain a significant difference with the interval case: the w-limit set can be locally connected, even connected.

References

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S. Agronsky, J. Ceder, What sets can be -limit sets in Eⁿ?, Real Analysis Exchange 17 (1991-92), 97-109.

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F. Balibrea, J.S. Cánovas, A. Linero, Minimal sets of antitriangular maps, submitted to International Journal of Bifurcation and Chaos.

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L. Block, W. A. Coppel, "One-Dimensional Dynamics", Lecture Notes in Mathematics 1513, Springer-Verlag, Berlin, Heidelberg (1992).

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J. Ceder, Some results and problems about w-limit sets, Real Analysis Exchange 16 (1990-91), 39-40.

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S. F. Kolyada, L. Snoha, On w-limit sets of triangular maps, Real Analysis Exchange, Vol. 18(1), (1992-93), 115-130.

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M. Kopel, Simple and Complex Adjustment Dynamics in Cournot Duopoly Models, Chaos, Solitons & Fractals, Vol. 7, No. 12 (1996), 2031-2048.

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T. Puu, "Nonlinear economic dynamics", Springer-Verlag, 1997.

Date received: February 27, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafw-37.