4/16/2023 0 Comments Sturmian subshift![]() ![]() Īvgustinovich, S., Karhumäki, J., Puzynina, S.: On abelian versions of critical factorization theorem. France 119, 199–215 (1991)Īvgustinovich, S., Cassaigne, J., Karhumäki, J., Puzynina, S., Saarela, A.: On abelian saturated infinite words. Adamczewski, B.: Balances for fixed points of primitive substitutions. Proposition: Suppose the subshift of finite type is represented by a matrix.Remark: Because is compact and is continuous in the above setting, it follows that is bounded to one (this means that ).Theorem 3: Given a sofic shift there exists an SFT and a factor map that is finite to one (this means that for all.Theorem 2 (Surjectivity is decidable): There is an algorithm that given a finite set and a function decides if it induces a surjective function by.Decidability of isomorphism is a notorious open question even for SFTs, and for sofic shifts this is a more general question. It is crucial that we do not ask if the two subshifts are isomorphic, but rather if they are equal.Theorem 1 (recognizing a sofic shiftis decidable): There exists an algorithm that decides if two labeled graphs represents the same sofic shift.Proposition: A subshift is sofic if and only if is a regular language.Proposition: Any sofic shift can be represented by a labeled directed graph as above.Representation of sofic shifts via labeled direct graph: A labeled directed multigraph is a triple, where is a directed multigraph, is a finite set, and is a function called the edge labeling. The set of bi-infinite labelings of paths in is a sofic shift.Example “The sunny-side-up shift”: This is the subshift consisting of the points with at most non-zero coordinate.Example: The even shift is sofic but not an SFT.An SFT extension of a sofic shift is sometimes called a Sofic shifts: (definition) A subshift is called sofic if it is a factor of an SFT.If is the same homeomorphism for every, this is a direct product. Example- Skew products: Let be a TDS, a compact (metric) topological space, a continuous function.In this case we say that the map is a factor map or semi-conjugacy, and that is an extension of. Factor maps: (definition) We say that a TDS is a factor of another TDS if there exists a map that is continuous, surjective, and equivariant.Exercise: An SFT is irreducible and aperiodic if and only if it is topologically mixing.Topologically mixing TDS: ( definition) A TDS is called topologically mixing if for every pair of non-empty open sets there exists so that for every.Aperiodic SFT: Represented by an essential aperiodic graph.Aperiodic graph: A directed graph is called aperiodic if the least common multiplier (LCM) of the lengths of directed cycles in is. ![]() An irriducible graph has an irriducible adjecncy matrix. Exercise: If $\latex M$ is not irreducible, by permuting the rows an columns it is an upper-diagonal block matrix.In this case we say that the matrix is irreducible. Exercise: A essential matrix represents an irreducible SFT if and only if for every there exists so that.Irreducible directed graphs: A directed graph is called (strongly) irreducible if there exists a directed path between any two vertices.(a TDS with a dense forward orbit is called topologically forward transitive). Exercise: A subshift is irreducible iff there is a point so that the forward orbit is dense.Irreducible subshifts: (definition) A subshift is called irreducible if for every there exists so that.Exercise: Given a matrix, show how to find an essential matrix that represents an isomorphic subshift.Similarly, a non-negative integer valued matrix is called essential if for every there exists so that and. Equivalently, for every vertex there exists a bi-infinite directed path that passes through the vertex. Essential graph: A directed graph is called essential if every vertex has at least one incoming edge and at least one outgoing edge.Exercise: If is sofic then is an algebriac integer (it is a root of of monic polyonomial with integer coefficients).Exercise: is a sofic shift iff is eventually periodic.Proposition: For every there exists a sequence so that and.From the above exercise it is not difficult to deduce the following:.Exercise: If and for all then for all.Bipartite codes : Suppose and $\mathcal$.Let’s start with an simple and general observation. Suppose and are compact metric spaces, and that are homeomorphisms between them.More specifically, given a pair of non-negative integer valued matrics, when do they represent isomorphic subshifts? Main problem: Given two SFTs, are they isomorphic (topologically conjugate).Follows Chapter 7 in the Lind-Marcus book. ![]()
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