Some continua admitting expansive homeomorphisms
by
Francisco Balibrea Gallego
Departamento de Matemáticas.- Universidad de Murcia

A continuum X is a nondegenerate compact, connected metric space. A homeomorphism h from X into itself is called expansive provided that for every x, y belonging to X, x different to y, there exists a positive constant c such that if

d(hn(x), hn(y)) <= c

for all integer then is x=y

Expansive homeomorphisms exhibit "chaotic behaviour" since no matter how two close points are, either their forward or reverse image will eventually be a certain distance apart. This is interesting from the dynamical point of view since the so called strange attractor admit expansive homeomorphisms which means a very complicated dynamics inside them.

Plykin´s attractors or the dyadic solenoid are also examples of this behaviour. But there are examples og other continua non admitting the expansive property. For example, rotations on the circle, translations on the torus, linear flows on the torus or the shift homeomorphism of the inverse limit of any continuous surjection of an arc.

The aim of this paper is to survey some examples of continua appearing in certain problems, not having the expansiveness property like dendroids and chainable continua, to prove that tree-like continua does not hold the property and show that Plykin´s attractor does it.

Some open problems are also presented.

Date received: March 12, 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-63.