zbMATH — the first resource for mathematics

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).

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
Full Text: DOI
[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.