Entropy for dynamical sources
Résumé
A quite general model of source that comes from dynamical systems theory is introduced. Within this model, basic problems of algorithmic information theory contexts are analysed. The main tool for the analysis is a (generalized) trans-fer operator, which can be viewed as a "generating" operator for "fundamental" probabilities. Its dominant spectral objects are related to important parameters of the source, such as the entropy rate, and play a central rôle in all the results. 1 Dynamical sources. In information theory contexts, data items are (infinite) words that are pro-duced by a common mechanism, called a source. Real-life sources are often complex objects. We introduce here a general framework of sources related to dynamical systems theory which can be viewed as a generalization of numer-ation systems. This model goes beyond the cases of memoryless and Markov chains, it can describe nonmarkovian processes, where the dependency on past history is unbounded, and as such, it attains a high level of generality. A probabilistic dynamical source is defined by five elements: (a) an alphabet Σ, finite or denumerable, (b) a topological partition of I := [0, 1] with disjoint open intervals (I m) m∈Σ , (c) an encoding mapping σ which is constant and equal to m on each I m , (d) a shift mapping T whose restriction to I m is a bijection of class C 2 from I m to J m := T (I m). The local inverse of T | Im is denoted by h m , (e) a real density f on interval I, and its associated distribution function F . The dynamical system S := (I, T), defined by the first four elements, asso-ciates to a real number x ∈ I the word M (x) := (σx, σT x, σT 2 x, .
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