Katok, A.; Spatzier, R. J. Invariant measures for higher-rank hyperbolic abelian actions. (English) Zbl 0859.58021 Ergodic Theory Dyn. Syst. 16, No. 4, 751-778 (1996); correction ibid. 18, No. 2, 503-507 (1998). The authors investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of \(\mathbb{R}^k\), \(\mathbb{Z}^k\) and \(\mathbb{Z}^k_+\). It is shown that they are either Haar measures or that every element of the action has zero metric entropy. A conjecture is established that this result may be fairly extended. Reviewer: J.Ombach (Kraków) Cited in 9 ReviewsCited in 49 Documents MSC: 37D99 Dynamical systems with hyperbolic behavior 37A99 Ergodic theory 54C70 Entropy in general topology 54H15 Transformation groups and semigroups (topological aspects) Keywords:actions of higher-rank Abelian groups on compact manifolds; partial hyperbolicity; Haar measure; ergodic invariant measures; metric entropy PDFBibTeX XMLCite \textit{A. Katok} and \textit{R. J. Spatzier}, Ergodic Theory Dyn. Syst. 16, No. 4, 751--778 (1996; Zbl 0859.58021) Full Text: DOI References: [1] Margulis, Discrete Subgroups of Lie Groups (1991) · doi:10.1007/978-3-642-51445-6 [2] DOI: 10.2307/2944357 · Zbl 0763.28012 · doi:10.2307/2944357 [3] DOI: 10.2307/1971328 · Zbl 0605.58028 · doi:10.2307/1971328 [4] Katok, Workshop on Lie Groups, Ergodic Theory and Geometry pp 36– (1992) [5] DOI: 10.1007/BF02808018 · Zbl 0790.28012 · doi:10.1007/BF02808018 [6] Huyi, Ergod. Th. & Dynam. Sys. 13 pp 73– (1993) [7] Host, Nombres normaux, entropie, translations · Zbl 0839.11030 · doi:10.1007/BF02761660 [8] DOI: 10.1007/BF01692494 · Zbl 0146.28502 · doi:10.1007/BF01692494 [9] DOI: 10.1007/BF02764832 · Zbl 0783.28009 · doi:10.1007/BF02764832 [10] Berend, Ergod. Th. & Dynam. Sys. 4 pp 499– (1984) [11] DOI: 10.2307/1999631 · Zbl 0532.10028 · doi:10.2307/1999631 [12] Anosov, Proc. Steklov Inst. Maths. 90 pp none– (1967) [13] Lang, Algebraic Number Theory (1970) · Zbl 0211.38404 [14] Lang, Algebra (1986) [15] Koppel, A.M.S. Proc. Symp. Pure Math. 14 pp 165– (1970) · doi:10.1090/pspum/014/0270396 [16] Koblitz, p-adic Numbers, p-adic Analysis, and Zeta Functions (1984) · doi:10.1007/978-1-4612-1112-9 [17] Katok, Differential rigidity of Anosov actions of higher-rank abelian groups and algebraic lattice actions (1996) · Zbl 0938.37010 [18] Katok, Math. Res. Lett. 1 pp 193– (1994) · Zbl 0836.57026 · doi:10.4310/MRL.1994.v1.n2.a7 [19] Katok, Publ. Math. IHES 79 pp 131– (1994) · Zbl 0819.58027 · doi:10.1007/BF02698888 [20] Katok, Pac. J. Math. 170 pp 105– (1995) · Zbl 0866.28015 · doi:10.2140/pjm.1995.170.105 [21] DOI: 10.1007/BF02776025 · Zbl 0785.22012 · doi:10.1007/BF02776025 [22] Katok, Ergod. Th. & Dynam. Sys. 15 pp 569– (1995) [23] Satayev, Usp. Math. Nauk 30 pp 203– (1975) [24] DOI: 10.2307/1997399 · Zbl 0329.58013 · doi:10.2307/1997399 [25] Rudolph, Ergod. Th. & Dynam. Sys. 10 pp 395– (1990) [26] Rohklin, A.M.S. Transl. (I) 10 pp 1– (1962) [27] Rees, Some (1982) [28] DOI: 10.1007/BF02766212 · Zbl 0654.28010 · doi:10.1007/BF02766212 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.