Decimation and interleaving operations in one-sided symbolic dynamics

Interleaving of path sets

“…The finiteness result in (3) also follows from (2). For general closed sets X, the paper [5] gave examples having an infinite depth tree of recursively refined iterated factorizations. The path set concept has previously been studied in automata theory and formal language theory, given in different terminology.…”

“…These closure operations were studied in [5]. One always has X ⊆ X [n] , the operation is idempotent:…”

“…The paper [5] gave a classification of the possible pattern of interleaving factorizations for a general set X, and a separate classification for a general closed set X ⊂ A N , see Section 6.1. For a closed set X ⊂ A N , exactly one of the following holds.…”

“…It gives examples showing the output presentations need not be right-resolving, even when the presentations of the P i are minimal rightresolving. (4) Section 5 first reviews results for decimation and interleaving operations acting on general sets X ⊂ A N which were shown in [5]. That paper defines a hierarchy of closure operations X → X [n] , the n-th interleaving closure of X, and shows that a set X is n-factorizable if and only if X…”

“…Decimation and interleaving operations are initially defined using individual elements of A N , extend to act on arbitrary subsets X ⊆ A N by set union, and then are specialized to path sets P ⊆ A N in this paper. The paper [5] studies these operations acting on general sets X ⊆ A N .…”