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Conditional entropy theory in infinite measure and a question of Krengel

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Abstract

We develop a conditional entropy theory for infinite measure preserving actions of countable discrete amenable groups with respect to a σ-finite factor. This includes ‘infinite’ analogues of relative Kolmogorov-Sinai, Rokhlin and Krieger theorems on generating partitions, Pinsker theorem on disjointness, Furstenberg decomposition and disjointness theorems, etc. In case of ℤ-action, our concept of relative entropy matches well the ‘absolute’ entropy h Kr introduced by Krengel. Answering in part his question and a question of Silva and Thieullen, we show that for any non-distal transformation S there exists an infinite measure preserving transformation T with h Kr(T × S) = ∞ but h Kr(T) = 0.

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Authors and Affiliations

  1. Institute for Low Temperature Physics & Engineering of Ukrainian National Academy of Sciences, 47 Lenin Ave., Kharkov, 61164, Ukraine

    Alexandre I. Danilenko

  2. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523-1874, USA

    Daniel J. Rudolph

Authors
  1. Alexandre I. Danilenko
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  2. Daniel J. Rudolph
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Correspondence to Alexandre I. Danilenko.

Additional information

This project was supported in part by a CRDF grant UM1-2546-KH-03.

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Danilenko, A.I., Rudolph, D.J. Conditional entropy theory in infinite measure and a question of Krengel. Isr. J. Math. 172, 93–117 (2009). https://doi.org/10.1007/s11856-009-0065-2

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  • DOI: https://doi.org/10.1007/s11856-009-0065-2

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