# Matrix problems arising from symbolic dynamics

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Matrix problems arising from symbolic dynamics Mike Boyle University of Maryland Kansas State University May 5 2009 Outline of the talk I 1 Shifts of finite type I 2 Strong shift equivalence I 3 SSE and SFTs I 4 Shift equivalence I 5 More on SSE as a matrix relation I 6 The Spectral Conjectures I 7 G SFTs II 1 Polynomial matrices II 2 Positive equivalence II 3 Positive K theory 1 III 1 Flow equivalence III 2 Flow equivalence of mixing SFTs III 3 Equivariant flow equivalence of G SFTs IV Flow equivalence of sofic shifts V References 2 I 1 Shifts of finite type Given A an n n matrix over Z view A as adjacency matrix of directed graph GA on vertices 1 2 n A i j number of edges from i to j Let XA be the space of doubly infinite sequences x x 1 x 0 x 1 such that for all n x n is an edge of GA and x n 1 follows x n in GA XA is naturally a compact metrizable space A XA XA is the shift homeomorphism A x n x n 1 3 The topological dynamical system A is a shift of finite type SFT It is a mixing SFT if the matrix A is primitive nonnegative with some An strictly positive The mixing SFTs analogous to primitive among nonnegative square matrices are the basic building blocks and the most important case of SFT Two top dyn systems S and T are topologically conjugate or isomorphic T S if there is some homeomorphism h such that hS T h Every SFT is isomorphic to some A SFTs play a significant role in dynamical systems To study topological conjugacy of SFTs A in terms of the defining matrices A we must define some matrix relations 4 I 2 Strong shift equivalence S a subset of a ring containing 0 and 1 Matrices A B are elementary strong shift equivalent over S ESSE S if there exist matrices U V over S such that A U V and B V U The relation strong shift equivalence over S SSE S is the transitive closure of ESSE S Example 1 A 2 1 1 U1V1 1 B 1 1 1 1 1 1 1 1 V1U1 1 1 1 1 1 0 B 0 1 1 0 0 1 1 1 1 0 1 0 1 1 C 0 1 1 0 1 0 0 1 1 1 1 1 0 1 U2V2 1 0 V2U2 1 5 So A and C are SSE Z A and C are not ESSE