In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. They also describe the set of all possible sequences executed by a finite state machine. The most widely studied shift spaces are the subshifts of finite type.
Let V be a finite set of n symbols (alphabet). Let X denote the set V^\Z of all bi-infinite sequences of elements of V together with the shift operator T. We endow V with the discrete topology and X with the product topology. A symbolic flow or subshift is a closed T-invariant subset Y of X and the associated language LY is the set of finite subsequences of Y.
Now let A be an n × n adjacency matrix with entries in {0, 1}. Using these elements we construct a directed graph G = (V, E) with V the set of vertices and E the set of edges containing the directed edge x → y in E if and only if A_x, y = 1. Let Y be the set of all infinite admissible sequences of edges, where by admissible it is meant that the sequence is a walk of the graph, and the sequence can be either one-sided or two-sided infinite. Let T be the left shift operator on such sequences; it plays the role of the time-evolution operator of the dynamical system. A subshift of finite type is then defined as a pair (Y, T) obtained in this way. If the sequence extends to infinity in only one direction, it is called a one-sided subshift of finite type, and if it is bilateral, it is called a two-sided subshift of finite type.
Formally, one may define the sequences of edges as
This is the space of all sequences of symbols such that the symbol p can be followed by the symbol q only if the (p, q)-th entry of the matrix A is 1. The space of all bi-infinite sequences is defined analogously:
The shift operator T maps a sequence in the one- or two-sided shift to another by shifting all symbols to the left, i.e.
Clearly this map is only invertible in the case of the two-sided shift.
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In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator. Formally, a Markov partition is used to provide a finite cover for the smooth system; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one covering set to another.
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process.
In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. In all usual cases, the largest eigenvalue is 1, and the corresponding eigenvector is the invariant measure of the system. The transfer operator is sometimes called the Ruelle operator, after David Ruelle, or the Perron–Frobenius operator or Ruelle–Perron–Frobenius operator, in reference to the applicability of the Perron–Frobenius theorem to the determination of the eigenvalues of the operator.
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