On extensions of dynamical systems by function algebras
by
F. Garibay-Bonales
Universidad Michoacana de San Nicolás de Hidalgo
Coauthors: M. Sanchis Lópes, R. Vera Mendoza

Let (X,   \tau) be a Tychonoff space, \Sigma the set of natural numbers or the integers, and j:X --> X a continuous function or a homeomorphism.

Given x in X, its orbit is the set O(x) = { j^k(x) |  k in \Sigma}. The dynamical system (X,  j,  \Sigma) is called topologically transitive if there exists x in X such that O(x) is a dense set. (X,  j,  \Sigma) is minimal if O(x) is dense for each x in X.

If (X,  j,  \Sigma) is topologically transitive (minimal), it is of interest to know when an extension (K,  [^(j)],  \Sigma) has the same property.

Pennings and Peters introduced extensions of dynamical systems by using function algebras. If B(X) is the set of all bounded complex-valued functions on X, with the norm of the supremun, and U is a C^*-algebra of functions of X with C(X) subset U subset B(X), they constructed an extension (K,  [^(j)],  \Sigma) where K was the spectrum of the algebra U. Under some restrictions on U we obtain transitivity and minimality of the extension.

Here we construct extensions of (X,  j,  \Sigma) as follows: We take a topology \tau^* finer than \tau, and a compactification K of (X,  \tau^*). X is a dense set of K and j: X --> X can be extended naturally to a continuous mapping [^(j)] : K --> K with [^(j)]|_X = j. In this way, (K,  [^(j)],  \Sigma) is an extension of (X,  j,  \Sigma) where the continuous surjection p: K --> X is such that p|_X = identity on X. We show that the extensions obtained in this manner are the same as the ones obtained by Pennings and Peters but the topological point of view allows us to improve their results on transitivity and minimality.

Date received: March 4, 1997

Copyright © 1997 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 # caak-56.