2014/05/20 by Tim Austin, Austin, Tim
Mathematics · Physics and Astronomy · #37A30 #37A35 (primary) #37A50 #60J05 (secondary) #Advanced Thermodynamics and Statistical Mechanics #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.1405.5121
openalex publication_date 2014/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A number of recent works have sought to generalize the Kolmogorov-Sinai entropy of probability-preserving transformations to the setting of Markov operators acting on the integrable functions on a probability space (X,μ). These have culminated in a proof by Downarovicz and Frej that these definitions all coincide, and that the resulting quantity is uniquely characterized by certain properties. On the other hand, Makarov has shown that this `operator entropy' is always dominated by the Kolmogorov-Sinai entropy of a classical system that may be constructed from a Markov operator, and that these numbers coincide under certain extra assumptions. This note proves that equality in all cases.