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Entropy of probability kernels from the backward tail boundary

Volume 227 / 2015

Tim Austin Studia Mathematica 227 (2015), 249-257 MSC: Primary 37A30, 37A35; Secondary 37A50, 60J05. DOI: 10.4064/sm227-3-4


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 $(X,\mu )$. These works have culminated in a proof by Downarowicz and Frej that various competing definitions all coincide, and that the resulting quantity is uniquely characterized by certain abstract properties.

On the other hand, Makarov has shown that this `operator entropy' is always dominated by the Kolmogorov–Sinai entropy of a certain 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.


  • Tim AustinCourant Institute of Mathematical Sciences
    New York University
    251 Mercer St
    New York, NY 10012, U.S.A.

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