Non-abelian free group actions: Markov processes, the Abramov-Rohlin formula and Yuzvinskii’s formula. (English) Zbl 1221.37009
Ergodic Theory Dyn. Syst. 30, No. 6, 1629-1663 (2010); corrigendum ibid. 33, No. 2, 643-645 (2013).
The author introduces Markov chains and processes over non-abelian free groups and semigroups. He proves a formula for the \(f\)-invariant of a Markov chain over a free group in terms of transition matrices and parallels the classical formula for the entropy of a Markov chain. Applications include free group analogues of the Abramov-Rohlin formula for skew-product actions and Yuzvinskii’s addition formula for algebraic actions.
Reviewer: Bolis Basit (Clayton)
MSC:
| 37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |
| 37B40 | Topological entropy |
References:
| [1] | Yuzvinskii, Izv. Akad. Nauk SSSR Ser. Mat. 29 pp 1295– (1965) |
| [2] | DOI: 10.1007/BF01231517 · Zbl 0774.22002 · doi:10.1007/BF01231517 |
| [3] | DOI: 10.1007/BF01299386 · Zbl 0764.28014 · doi:10.1007/BF01299386 |
| [4] | Thomas, Trans. Amer. Math. Soc. 160 pp 119– (1971) |
| [5] | Kolmogorov, Dokl. Akad. Nauk SSSR 124 pp 754– (1959) |
| [6] | Kolmogorov, Dokl. Akad. Nauk SSSR (N.S.) 119 pp 861– (1958) |
| [7] | Katok, J. Mod. Dyn. 1 pp 545– (2007) · Zbl 1149.37001 · doi:10.3934/jmd.2007.1.545 |
| [8] | Glasner, Ergodic Theory Via Joinings pp xii+384– (2003) · Zbl 1038.37002 · doi:10.1090/surv/101 |
| [9] | DOI: 10.1016/0001-8708(70)90010-1 · Zbl 0203.05801 · doi:10.1016/0001-8708(70)90010-1 |
| [10] | DOI: 10.1112/S0024609399006104 · Zbl 1017.37007 · doi:10.1112/S0024609399006104 |
| [11] | DOI: 10.1017/S0143385706000939 · Zbl 1128.22003 · doi:10.1017/S0143385706000939 |
| [12] | DOI: 10.1023/B:GEOM.0000006580.47816.e9 · Zbl 1074.52007 · doi:10.1023/B:GEOM.0000006580.47816.e9 |
| [13] | DOI: 10.1090/S0894-0347-06-00519-4 · Zbl 1104.22010 · doi:10.1090/S0894-0347-06-00519-4 |
| [14] | DOI: 10.1016/0095-8956(86)90029-8 · Zbl 0602.05050 · doi:10.1016/0095-8956(86)90029-8 |
| [15] | DOI: 10.1007/s006050170003 · Zbl 0996.37007 · doi:10.1007/s006050170003 |
| [16] | Benjamini, Ann. Probab. 29 pp 1– (2001) |
| [17] | DOI: 10.3934/dcds.2009.25.981 · Zbl 1179.37012 · doi:10.3934/dcds.2009.25.981 |
| [18] | DOI: 10.1007/BFb0097526 · doi:10.1007/BFb0097526 |
| [19] | Abramov, Vestnik Leningrad. Univ. Math. 17 pp 5– (1962) |
| [20] | Sinaĭ, Dokl. Akad. Nauk SSSR 124 pp 768– (1959) |
| [21] | DOI: 10.1214/ss/1177010038 · Zbl 0955.60528 · doi:10.1214/ss/1177010038 |
| [22] | DOI: 10.1214/aop/1176990223 · Zbl 0758.60010 · doi:10.1214/aop/1176990223 |
| [23] | Parry, Entropy and Generators in Ergodic Theory pp xii+124– (1969) |
| [24] | DOI: 10.1007/BF02790325 · Zbl 0637.28015 · doi:10.1007/BF02790325 |
| [25] | DOI: 10.1016/0001-8708(70)90008-3 · Zbl 0227.28014 · doi:10.1016/0001-8708(70)90008-3 |
| [26] | DOI: 10.1016/0001-8708(70)90029-0 · Zbl 0197.33502 · doi:10.1016/0001-8708(70)90029-0 |
| [27] | McKay, European J. Combin. 4 pp 149– (1983) · Zbl 0517.05043 · doi:10.1016/S0195-6698(83)80045-6 |
| [28] | DOI: 10.1017/S096354830500684X · Zbl 1076.05007 · doi:10.1017/S096354830500684X |
| [29] | Wilson, Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996) pp 296– (1996) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
