![]() ![]() The Markov measure of a cylinder set may then be defined by μ ( C t ) = π a 0 p a 0, a 1 ⋯ p a s − 1, a s The stationary probability vector π = ( π i ) has all π i ≥ 0, ∑ π i = 1 and has ∑ i = 1 n π i p i j = π j.Ī Markov chain, as defined above, is said to be compatible with the shift of finite type if p i j = 0 whenever A i j = 0. A common object of study is the Markov measure, which is an extension of a Markov chain to the topology of the shift.Ī Markov chain is a pair ( P,π) consisting of the transition matrix, an n × n matrix P = ( p i j ) for which all p i j ≥ 0 and ∑ j = 1 n p i j = 1įor all i. MeasureĪ subshift of finite type may be endowed with any one of several different measures, thus leading to a measure-preserving dynamical system. In fact, both the one- and two-sided shift spaces are compact metric spaces. One can define a metric on a shift space by considering two points to be "close" if they have many initial symbols in common this is the p-adic metric. ![]() MetricĪ variety of different metrics can be defined on a shift space. ![]() In particular, the shift T is a homeomorphism that is, with respect to this topology, it is continuous with continuous inverse. Every open set in the subshift of finite type is a countable union of cylinder sets. Now let A be an n × n adjacency matrix with entries in A symbolic flow or subshift is a closed T-invariant subset Y of X and the associated language L Y is the set of finite subsequences of Y. We endow V with the discrete topology and X with the product topology. Let X denote the set V Z of all bi-infinite sequences of elements of V together with the shift operator T. Let V be a finite set of n symbols (alphabet). ![]()
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