On the Transitive Operators
by
Vladimir Todorov
Higher Institute for architecture and Civil Engineering, Sofia, Bulgaria

In this paper we investigate the transitivity of the so called shift operators. Let X be a topological space and Y subset X. The map f : Y --> X is said to be transitive if there exists an element x₀ in Y for which the forward orbit O₊f(x₀)={ fⁿ(x₀) | n = 1, 2, ... } is a dense subset of X.

Furthermore we need the following denotations:

• \nu: N --> N; one-to-one correspondence between the set of the naturals which satisfies the property:
  

  \infty Ç n=1  \nun(N) = \emptyset

• A sequence of separable topological spaces X_n and their subspaces Y_n subset Y_n
• A fixed map f_n:Y_\nu(n) --> X_n which is önto" for every n in \N.

Using the above notions for the spaces Y = \prod_n=1^\infty Y_n and X = \prod_n=1^\infty X_n we define a shift operator f : Y --> X by the expression:

f(x) = (f1(x(\nu(1))), f2(x(\nu(2))), ..., fn(x(\nu(n))), ...)

The main result of this paper states that every shift operator is transitive. Different corollaries are obtained - for example for linear operators in the separable Frechet spaces, the Keller spaces as well in some other types of products of spaces.

Date received: February 26, 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 # caam-19.