Symbolic dynamics from signed matrices

Jim Wiseman
2004 Discrete and Continuous Dynamical Systems. Series A  
We consider a method for assigning a sofic shift to a (not necessarily nonnegative integer) matrix by associating to it a directed graph with some vertices labelled 1 and the rest 2 (the decomposition of the vertices is arbitrary -in applications the choice should be natural). We can detect positive topological entropy for this sofic shift by comparing the characteristic polynomial of the original matrix to those for the matrices for the restrictions of the shifts to each piece (1 and 2). Our
more » ... in application is to the use of the Conley index to detect symbolic dynamics in isolated invariant sets, and is an extension of a result by Carbinatto, Kwapisz, and Mischaikow. Example 3.2. Let M = 1 3 −2 0 , corresponding to the graph in Figure 2 (the number along each edge represents its weight). then there must be an M -path of length m from v i to v j . The converse is not true, as the following example shows. (It is true if M is nonnegative.) Example 3.3. Let M = 1 −1 1 −1 . Then for any m ≥ 1 and any i and j, there is an M -path of length m from v i to v j . But M m = 0 for m ≥ 2. Next, we see that there is a strong relation between M -loops of a matrix M and its characteristic polynomial. Proposition 3.4. For 1 ≤ i, j ≤ n and m ≥ 1, (M m ) ji = w(γ), where the sum is taken over all M -paths from v i to v j of length m. Proof. First, observe that where the last sum is taken over all k such that there is a (necessarily unique) path γ k from v i to v k of length one. The result then follows from linearity and the fact that every path is a concatenation of paths of length one. (Compare [9, Lemma III.2.2], which is the same proposition for nonnegative adjacency matrices.) Corollary 3.5. For 1 ≤ i ≤ n and m ≥ 1, (M m ) ii = w(λ), where the sum is taken over all M -loops at v i of length m. Corollary 3.6. The characteristic polynomial of M depends only on the set of M -loops λ with 1 ≤ l(λ) ≤ n and their weights w(λ). Proof. Newton's formula ( [4, §92]) tells us that knowledge of tr(M m ) for 1 ≤ m ≤ n is equivalent to knowledge of the characteristic polynomial of M . Since tr(M m ) = n i=1 (M m ) ii , the result follows from Corollary 3.5.
doi:10.3934/dcds.2004.11.621 fatcat:gotabqg6tfgvvkubsgdjmz7hnu