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A Juzvinskii addition theorem for finitely generated free group actions

Published online by Cambridge University Press:  01 November 2012

LEWIS BOWEN  and
YONATAN GUTMAN
LEWIS BOWEN
Affiliation:
Mathematics Department, Mailstop 3368, Texas A&M University, College Station, TX 77843-3368, USA (email: [email protected])
YONATAN GUTMAN
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland (email: [email protected])
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Abstract

The classical Juzvinskii addition theorem states that the entropy of an automorphism of a compact group decomposes along invariant subgroups. Thomas generalized the theorem to a skew-product setting. Using L. Bowen’s f-invariant, we prove the addition theorem for actions of finitely generated free groups on skew-products with compact totally disconnected groups or compact Lie groups (correcting an error in L. Bowen [Nonabelian free group actions: Markov processes, the Abramov–Rohlin formula and Yuzvinskii’s formula. Ergod. Th. & Dynam. Sys.30(6) (2010), 1629–1663]) and discuss examples.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 1 , February 2014 , pp. 95 - 109
Copyright
Copyright © 2012 Cambridge University Press 

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References

[BG12]Bowen, L. and Gutman, Y.. Corrigendum to Nonabelian free group actions: Markov processes, the Abramov–Rohlin formula and Yuzvinskii’s formula. Ergod. Th. & Dynam. Sys. (2012) to appear, http://arxiv.org/abs/1206.6772.Google Scholar
[BM09]Björklund, M. and Miles, R.. Entropy range problems and actions of locally normal groups. Discrete Contin. Dyn. Syst. 25(3) (2009), 981–989.Google Scholar
[Bo10a]Bowen, L.. A measure-conjugacy invariant for actions of free groups. Ann. of Math. (2) 171(2) (2010), 1387–1400.Google Scholar
[Bo10b]Bowen, L.. Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23 (2010), 217–245.Google Scholar
[Bo10c]Bowen, L.. Nonabelian free group actions: Markov processes, the Abramov–Rohlin formula and Yuzvinskii’s formula. Ergod. Th. & Dynam. Sys. 30(6) (2010), 1629–1663.Google Scholar
[He01]Helgason, S.. Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics, 34). American Mathematical Society, Providence, RI, 2001, xxvi+641 pp. Corrected reprint of the 1978 original.CrossRefGoogle Scholar
[Ju65]Juzvinskii, S. A.. Metric properties of the endomorphisms of compact groups. Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 1295–1328.Google Scholar
[KH95]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995, 822 pp.Google Scholar
[Ku65]Kushnirenko, A. G.. An upper bound of the entropy of classical dynamical systems. Sov. Math. Dokl. 6 (1965), 360–362.Google Scholar
[Li11]Li, H.. Compact group automorphisms, addition formulas and Fuglede–Kadison determinants. Ann. of Math. (2) 176(1) (2012), 303–347.Google Scholar
[LS09]Lind, D. and Schmidt, K.. Principal algebraic actions of discrete amenable groups and noncommutative Mahler measure. Preprint, 2009.Google Scholar
[LSW90]Lind, D., Schmidt, K. and Ward, T.. Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math. 101(3) (1990), 593–629.CrossRefGoogle Scholar
[MRV11]Meesschaert, N., Raum, S. and Vaes, S.. Stable orbit equivalence of Bernoulli actions of free groups and isomorphism of some of their factor actions. Expo. Math. to appear. Preprint, arXiv:1107.1357.Google Scholar
[Mi08]Miles, R.. The entropy of algebraic actions of countable torsion-free abelian groups. Fund. Math. 201 (2008), 261–282.Google Scholar
[OW87]Ornstein, D. and Weiss, B.. Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 (1987), 1–141.Google Scholar
[Th71]Thomas, R. K.. The addition theorem for the entropy of transformations of $G$-spaces. Trans. Amer. Math. Soc. 160 (1971), 119–130.Google Scholar