Mixing actions of the Heisenberg group
The paper is strongly motivated by the following question of Dan Rudolph: \textit{which amenable locally compact second countable groups admit mixing free actions with zero entropy?} This question is still (widely) open but the existence of an action (and of the group) with the above properties would follow from the existence of mixing rank-one free actions.NEWLINENEWLINEPreviously, the existence of mixing rank-one actions has been established for the groups \(\mathbb{Z}\) (in [\textit{T. M. Adams}, Proc. Am. Math. Soc. 126, No. 3, 739--744 (1998; Zbl 0906.28006); \textit{D. Creutz} and \textit{C. E. Silva}, Ergodic Theory Dyn. Syst. 24, No. 2, 407--440 (2004; Zbl 1066.37003); \textit{D. S. Ornstein}, in: Proc. 6th Berkeley Sympos. math. Statist. Probab., Univ. Calif. 1970, 2, 347--356 (1972; Zbl 0262.28009) and \textit{V. V. Ryzhikov}, J. Dyn. Control Syst. 3, No. 1, 111--127 (1997; Zbl 0995.37002)]), \(\mathbb{Z}^2\) (in [\textit{T. Adams} and \textit{C. E. Silva}, Ergodic Theory Dyn. Syst. 19, No. 4, 837--850 (1999; Zbl 0939.28013)]), \(\mathbb{R}\) (in [\textit{B. Fayad}, Invent. Math. 160, No. 2, 305--340 (2005; Zbl 1064.37006) and \textit{A. A. Prikhod'ko}, Sb. Math. 192, No. 12, 1799--1828 (2001); translation from Mat. Sb. 192, No. 12, 61--92 (2001; Zbl 1017.37002)]), and more generally for \(\mathbb{R}^{d_1}\times \mathbb{Z}^{d_2}\), for arbitrary \(d_1, d_2\geq 0\) (in [the author and \textit{C. E. Silva}, Ann. Inst. Henri Poincaré, Probab. Stat. 43, No. 4, 375--398 (2007; Zbl 1126.37004)]), as well as for direct sums of countably many finite groups (in [the author, Isr. J. Math. 156, 341--358 (2006; Zbl 1131.37006)]).NEWLINENEWLINEThe major result of the paper is to prove that the Heisenberg group \(H_3(\mathbb{R})\) also belongs to the above list. This is shown by constructing a rank-one action of \(H_3(\mathbb{R})\) which is mixing at all orders.NEWLINENEWLINEThe construction is fairly explicit utilizing the idea of the so-called \((C,F)\)-construction; the latter is a sort of generalization of the well-known cutting-and-stacking construction for the actions of \(\mathbb{Z}\). The author first shows that the constructed action is mixing when restricted to the center of \(H_3(\mathbb{R})\). Then, adapting ideas from [\textit{V. V. Ryzhikov}, Russ. Math. Surv. 49, No. 2, 170--171 (1994); translation from Usp. Mat. Nauk 49, No. 2(296), 163--164 (1994; Zbl 0836.28007)], it is shown that the entire action is mixing. The proof uses subtle analytical tools involving the Haar measure of \(H_3(\mathbb{R})\).NEWLINENEWLINEAs an application of the main results of the paper, the author shows that \(H_3(\mathbb{R})\) admits mixing Poisson and Gaussian actions.NEWLINENEWLINEThe paper ends with a list of interesting open problems.
scientific article; zbMATH DE number 6359407
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