Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T03:05:36.507Z Has data issue: false hasContentIssue false

A Juzvinskii addition theorem for finitely generated free group actions

Published online by Cambridge University Press:  01 November 2012

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])

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
Copyright
Copyright © 2012 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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), 981989.Google Scholar
[Bo10a]Bowen, L.. A measure-conjugacy invariant for actions of free groups. Ann. of Math. (2) 171(2) (2010), 13871400.Google Scholar
[Bo10b]Bowen, L.. Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23 (2010), 217245.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), 16291663.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), 12951328.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), 360362.Google Scholar
[Li11]Li, H.. Compact group automorphisms, addition formulas and Fuglede–Kadison determinants. Ann. of Math. (2) 176(1) (2012), 303347.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), 593629.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), 261282.Google Scholar
[OW87]Ornstein, D. and Weiss, B.. Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 (1987), 1141.Google Scholar
[Th71]Thomas, R. K.. The addition theorem for the entropy of transformations of $G$-spaces. Trans. Amer. Math. Soc. 160 (1971), 119130.Google Scholar