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Conjugacy classes are dense in the space of mixing $$\mathbb{Z}^d$$-actions. (English) Zbl 1376.37009
The main result of the paper is the extension of Halmos’ conjugacy lemma on the space of mixing $${\mathbb{Z}}^d$$-actions, namely, it proves the density of the conjugacy class of any mixing $${\mathbb{Z}}^d$$-action.
This implies that the set of $${\mathbb{Z}}^d$$-actions of rank 1 is massive, and the rank 1 property is generic for mixing $${\mathbb{Z}}^d$$-actions. This ensures the genericity for mixing of consequences of rank 1, such as triviality of the centralizer and the absence of factors.
The proof is based on an auxiliary result: “For any mixing $${\mathbb{Z}}^d$$-action $$T$$ and any numbers $$\alpha \in (0,1)$$ and $$\epsilon >0$$, there exists a set $$A$$ of measure $$\alpha$$ whose images $$\{T^g A\}_{g\in {\mathbb{Z}}^d}$$ intersect almost independently up to $$\epsilon$$”, which is formalized as a separate theorem (in a somewhat more general form).

MSC:
 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 37A25 Ergodicity, mixing, rates of mixing 37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems 22D40 Ergodic theory on groups
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References:
 [1] Halmos, P.R., In general a measure preservingtransformation ismixing, Ann.Math., 45, 786-792, (1944) · Zbl 0063.01889 [2] Foreman, M.; Weiss, B., An anti-classification theorem for ergodic measure preserving transformations, J. Eur.Math. Soc., 6, 277-292, (2004) · Zbl 1063.37004 [3] Alpern, St., Conjecture: in general a mixing transformation is not two-fold mixing, Ann. of Prob., 13, 310-313, (1985) · Zbl 0574.28012 [4] Tikhonov, S.V., Completemetric on the set ofmixing transformations, UspekhiMat. Nauk, 62, 209-210, (2007) [5] Tikhonov, S. V., A complete metric in the set of mixing transformations, Matem. Sb., 198, 135-158, (2007) · Zbl 1140.37005 [6] Tikhonov, S. V., A note on rochlin’s property in the space of mixing transformations, Mat. Zametki, 90, 953-954, (2011) · Zbl 1317.37008 [7] Tikhonov, S. V., Approximation of mixing transformations, Mat. Zametki, 95, 282-299, (2014) · Zbl 1370.37004 [8] Tikhonov, S. V., Mixing transformations with homogeneous spectrum, Matem. Sb., 202, 139-160, (2011) · Zbl 1247.37008 [9] Bashtanov, A. I., Generic mixing transformations are rank 1, Mat. Zametki, 93, 163-171, (2013) · Zbl 1312.37006 [10] Tikhonov, S. V., Genericity of a multiple mixing, UspekhiMat. Nauk, 67, 187-188, (2012) · Zbl 1284.37005 [11] Tikhonov, S. V., Complete metric on mixing actions of general groups, J. Dyn. Control Syst., 19, 17-31, (2013) · Zbl 1260.37004 [12] Ryzhikov, V. V., Pairwise ε-independence of the sets T ia for a mixing transformation T, Funktional. Anal. Prilozhen., 43, 88-91, (2009) · Zbl 1271.37004 [13] Ornstein, D., Bernoulli shifts with the same entropy are isomorphic, Adv. Math., 4, 337-352, (1970) · Zbl 0197.33502 [14] Sinai, Ya. G., General ergodic theory of transformation groups with invariant measure, in Current Problems in Mathematics: Dynamical Systems, 2, 5-111, (1985) [15] Kirillov, A. A., Dynamical systems, factors and representations ofgroups, UspekhiMat.Nauk, 22, 67-80, (1967) [16] Bashtanov, A. I., Property of almost independent images for ergodic transformations without partial rigidity, Trudy Mat. Inst. Steklov, 271, 29-39, (2010) · Zbl 1302.37003 [17] Katok, A. B., Entropy and approximations of dynamical systems by periodic transformations, Funktional. Anal. Prilozhen., 1, 75-85, (1967) · Zbl 0186.46801 [18] Katznelson, Y.; Weiss, B., Commuting measure-preserving transformations, Israel J. Math., 12, 161-173, (1972) · Zbl 0239.28014 [19] Ornstein, D.; Weiss, B., Ergodic theory of amenable group actions. I: the rohlin lemma, Bull. Amer. Math. Soc. (New Series), 2, 161-164, (1980) · Zbl 0427.28018 [20] Janvresse, E.; de la Rue, T.; Ryzhikov, V., Around king’s rank-one theorems: flows and zn-actions, Contemp. Mathematics, 567, 143-161, (2012) · Zbl 1303.37001 [21] Ryzhikov, V. V., On the asymmetry of multiple asymptotic properties of ergodic actions, Mat. Zametki, 96, 432-439, (2014) · Zbl 1370.37008
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