Information Meaning of Entropy of Nonergodic Measures

• Published in: Differential equations Vol. 55; no. 3; pp. 294 - 302
• Main Author: Bakhtin, V. I.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.03.2019; Springer
• Subjects: Difference and Functional Equations; Mathematics; Mathematics and Statistics; Ordinary Differential Equation; Ordinary Differential Equations; Partial Differential Equations
• ISSN: 0012-2661; 1608-3083
• DOI: 10.1134/S0012266119030029

The limit frequency properties of trajectories of the simplest dynamical system generated by the left shift on the space of sequences of letters from a finite alphabet are studied. The following modification of the Shannon-McMillan-Breiman theorem is proved: for any invariant (not necessarily ergodic) probability measure μ on the sequence space, the logarithm of the cardinality of the set of all μ -typical sequences of length n is equivalent to nh ( μ ), where h ( μ ) is the entropy of the measure μ . Here a typical finite sequence of letters is understood as a sequence such that the empirical measure generated by it is close to μ (in the weak topology).