Extensions of Dynamical Systems by Function Algebras
by
Manuel Sanchis
Universitat Jaume I
Coauthors: Juan J. Font

In [1] and [2], Peters and Pennings introduce the following kind of extensions of dynamical systems: Let X be compact Hausdorff, \Sigma the natural numbers or integers. Let j:X --> X be a continuous onto mapping. Then (X, j, \Sigma) is a dynamical system. Let U be a \Sigma-invariant C^*-algebra of bounded functions containing C(X). There is a natural extension ([^X], [^(j)], \Sigma) of (X, j, \Sigma) where [^(j)]([^x])(f) = 3D[^x] = (f o j), f in U, and [^X] is the structure space of U. They prove that, under certain restrictions on U, these extensions inherit some properties of (X, j, \Sigma) such as minimality, topological transitivity, sensitive dependence on initial conditions, ... In this paper, we continue their study by searching for sufficient conditions under which these extensions preserve recurrence, periodicity, almost periodicity, etc. We also provide some examples which show that our hypothesis cannot, in general, be relaxed and that some ergodic properties are not preserved.

[1] T. Pennings and J. Peters, Dynamical systems from function algebras , Proc. A.M.S. 105 (1989), 80-86.

[2] J. Peters and T. Pennings, Chaotic extensions of dynamical systems by function algebras, J. Math. Anal. Appl. 159 (1991), 345-360.

Date received: December 30, 1995

Copyright © 1995 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 # caad-03.