Disjointness preserving shifts on C0(X)

We study disjointness preserving (quasi-)n-shift operators on C0(X), where X is locally compact and Hausdorff. When C0(X) admits a quasi-n-shift T, there is a countable subset of X∞=X∪{∞} equipped with a tree-like structure, called φ-tree, with exactly n joints such that the action of T on C0(X) can be implemented as a shift on the φ-tree. If T is an n-shift, then the φ-tree is dense in X and thus X is separable. By analyzing the structure of the φ-tree, we show that every (quasi-)n-shift on c0 can always be written as a product of n (quasi-)1-shifts. Although it is not the case for general C0(X) as shown by our counter examples, we can do so after dilation.