IJPAM: Volume 68, No. 1 (2011)

TOPOLOGICAL DYNAMICS OF GENERIC PIECEWISE
CONTRACTIVE MAPS IN $n$ DIMENSIONS

Eleonora Catsigeras$^1$, Ruben Budelli$^2$
$^1$Institute of Mathematics
Faculty of Engineering
República University
565, Herrera y Reissig, Montevideo, 11300, URUGUAY
e-mail: [email protected]
$^2$Centre of Mathematics
Department of Biomathematics
Faculty of Sciences
República University
4225, Iguá, Montevideo, URUGUAY
e-mail: [email protected]


Abstract. We study the topological dynamics by iterations of a piecewise continuous, non linear and locally contractive map in a real finite dimensional compact ball. We consider those maps satisfying the ``separation property": different continuity pieces have disjoint images. The continuity pieces act as stable topological manifolds while the points in the discontinuity lines, separating different continuity pieces, act as topological saddles with an infinite expanding rate. We prove that $C^0$ generical systems exhibit one and at most a finite number of persistent periodic sinks attracting all the orbits. In other words, the chaotic behaviors that this class of mappings may exhibit, are structurally unstable and bifurcating.

Received: January 16, 2011

AMS Subject Classification: 37B25, 34C25, 37G15

Key Words and Phrases: piecewise continuous maps, periodic attractors, topological dynamics


Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2011
Volume: 68
Issue: 1