In the case that the system exhibits strong mixing properties, these bounds are explicitly expressed in terms of the speed of mixing. In this paper we provide general estimates for the errors between the distribution of the first, and more generally, the Kth return time (suitably rescaled) and the Poisson law for measurable dynamical systems. 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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