On some generic classes of ergodic measure preserving transformations

E. Glasner ; J.-P. Thouvenot ; B. Weiss

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 82 (2021) no. 1, pp. 19-44.

Résumé

We answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic transformations and their interrelations. In the second part of the paper we show that for a fixed ergodic $T$ with property $\mathbf{A}$, a generic extension $\widehat{T}$ of $T$ also has the property $\mathbf{A}$. Here $\mathbf{A}$ stands for each of the following properties: (i) having the same entropy as $T$, (ii) Bernoulli, (iii) K, and (iv) loosely Bernoulli. References: 46 entries.

Bibliographie

[1] R. L. Adler, P. C. Shields, “Skew products of Bernoulli shifts with rotations”, Israel J. Math., 12 (1972), 215–222 | DOI | MR | Zbl

[2] O. N. Ageev, “The generic automorphism of a Lebesgue space conjugate to a $G$-extension for any finite abelian group $G$”, Dokl. Akad. Nauk, 374:4 (2000), 439–442 (Russian) | MR | Zbl

[3] O N. Ageev, “On the genericity of some nonasymptotic dynamic properties”, Russian Math. Surveys, 58:1 (2003), 173–174 | DOI | MR | Zbl

[4] O. N. Ageev, “A typical dynamical system is not simple or semisimple”, Erg. Th. Dynam. Systems, 23 (2003), 1625–1636 | DOI | MR | Zbl

[5] T. Austin, “Measure concentration and the weak Pinsker property”, Publ. Math. IHES, 128 (2018), 1–119 | DOI | MR | Zbl

[6] J. Chaika, B. Kra, A prime system with many self-joinings, arXiv: 1902.02421v2 | MR

[7] A. del Junco, “A simple map with no prime factors”, Israel J. Math., 104 (1998), 301–320 | DOI | MR | Zbl

[8] A. del Junco, D. Rudolph, “On ergodic actions whose self-joinings are graphs”, Erg. Th. Dynam. Systems, 7 (1987), 531–557 | DOI | MR | Zbl

[9] A. del Junco, D. Rudolph, “Residual behavior of induced maps”, Israel J. Math., 93 (1996), 387–398 | DOI | MR | Zbl

[10] A. del Junco, M. Lemańczyk, M. K. Mentzen, “Semisimplicity, joinings and group extensions”, Studia Math., 112:2 (1995), 141–164 | MR | Zbl

[11] A. I. Danilenko, “On simplicity concepts for ergodic actions”, J. Anal. Math., 102 (2007), 77–117 | DOI | MR | Zbl

[12] T. de la Rue, J. de Sam Lazaro, “Une transformation générique peut être insérée dans un flot”, Ann. Inst. H. Poincaré Probab. Statist., 39:1 (2003), 121–134 | DOI | MR | Zbl

[13] N. A. Friedman, “Partial mixing, partial rigidity, and factors”, Measure and measurable dynamics (Rochester, NY, 1987), Contemp. Math., 94, AMS, Providence, RI, 1989, 141–145 | DOI | MR

[14] E. Glasner, “A topological version of a theorem of Veech and almost simple flows”, Erg. Th. Dynam. Systems, 10 (1990), 463–482 | DOI | MR | Zbl

[15] E. Glasner, Ergodic theory via joinings, Math. Surveys and Monographs, 101, AMS, Providence, RI, 2003 | DOI | MR | Zbl

[16] E. Glasner, B. Host, D. Rudolph, “Simple systems and their higher order self-joinings”, Israel J. Math., 78:1 (1992), 131–142 | DOI | MR | Zbl

[17] E. Glasner, J. L. King, “A zero-one law for dynamical properties”, Topological dynamics and applications (Minneapolis, MN, 1995), Contemp. Math., 215, AMS, Providence, RI, 1998, 231–242 | DOI | MR

[18] E. Glasner, B. Weiss, “A weakly mixing upside-down tower of isometric extensions”, Erg. Th. Dynam. Systems, 1 (1981), 151–157 | DOI | MR | Zbl

[19] E. Glasner, B. Weiss, “Relative weak mixing is generic”, Sci. China Math., 62:1 (2019), 69–72 | DOI | MR | Zbl

[20] P. R. Halmos, Lectures on ergodic theory, Publications of the Mathematical Society of Japan, 3, The Mathematical Society of Japan, 1956 | MR | Zbl

[21] R. Jones, W. Parry, “Compact abelian group extensions of dynamical systems. II”, Compositio Mathematica, 25:2 (1972), 135–147 | MR | Zbl

[22] A. Katok, A. Stepin, “Approximations in ergodic theory”, Russian Math. Surveys, 22:5 (1967), 77–102 | DOI | MR

[23] J. L. F. King, “Ergodic properties where order 4 implies infinite order”, Israel. J. Math., 80:1-2 (1992), 65–86 | DOI | MR | Zbl

[24] J. L. F. King, “The generic transformation has roots of all orders”, Colloq. Math., 84/85:2 (2000), 521–547 | DOI | MR | Zbl

[25] I. Meilijson, “Mixing properties of a class of skew-products”, Israel J. Math., 19 (1974), 266–270 | DOI | MR

[26] J. Melleray, “Extensions of generic measure-preserving actions”, Ann. Inst. Fourier, 64:2 (2014), 607–623 | DOI | MR | Zbl

[27] J. Melleray, T. Tsankov, “Generic representations of abelian groups and extreme amenability”, Israel J. Math., 198:1 (2013), 129–167 | DOI | MR | Zbl

[28] D. S. Ornstein, “A $K$ automorphism with no square root and Pinsker's conjecture”, Adv. Math., 10 (1973), 89–102 | DOI | MR | Zbl

[29] D. S. Ornstein, D. J. Rudolph, B. Weiss, Equivalence of measure preserving transformations, Mem. Amer. Math. Soc., 37, no. 262, 1982 | MR

[30] J. C. Oxtoby, Measure and category, Graduate Texts in Mathematics, 2, Springer–Verlag, New York–Berlin, 1980 | DOI | MR | Zbl

[31] W. Parry, “Ergodic properties of affine transformations and flows on nilmanifolds”, Amer. J. Math., 91 (1969), 757–771 | DOI | MR | Zbl

[32] D. J. Rudolph, “An example of a measure preserving map with minimal self-joinings, and applications”, J. Analyse Math., 35 (1979), 97–122 | DOI | MR | Zbl

[33] D. J. Rudolph, “Classifying the isometric extensions of a Bernoulli shift”, J. Analyse Math., 34 (1978), 36–60 | DOI | MR | Zbl

[34] D. J. Rudolph, “$k$-fold mixing lifts to weakly mixing isometric extensions”, Erg. Th. Dynam. Systems, 5:3 (1985), 445–447 | DOI | MR | Zbl

[35] D. J. Rudolph, “$\mathbb{Z}^n$ and $\mathbb{R}^n$ cocycle extensions and complementary algebras”, Erg. Th. Dynam. Systems, 6:4 (1986), 583–599 | DOI | MR | Zbl

[36] V. V. Ryzhikov, “Around simple dynamical systems. Induced joinings and multiple mixing”, J. Dynam. Control Systems, 3:1 (1997), 111–127 | DOI | MR | Zbl

[37] V. V. Ryzhikov, “Factors, rank, and embedding of a generic $\mathbb{Z}^n$-action in an $\mathbb{R}^n$-flow”, Russian Math. Surveys, 61:4 (2006), 786–787 | DOI | MR | Zbl

[38] V. V. Ryzhikov, J.-P. Thouvenot, “Disjointness, divisibility, and quasi-simplicity of measure-preserving actions”, Funct. Anal. Appl., 40:3 (2006), 237–240 | DOI | MR | Zbl

[39] M. Schnurr, “Generic properties of extensions”, Erg. Th. Dynam. Systems, 39 (2019), 3144–3168 | DOI | MR | Zbl

[40] S. Solecki, “Closed subgroups generated by generic measure automorphisms”, Erg. Th. Dynam. Systems, 34 (2014), 1011–1017 | DOI | MR | Zbl

[41] A. M. Stepin, “Spectral properties of typical dynamic systems”, Math USSR Izv., 29:1 (1987), 159–192 | DOI | MR | Zbl

[42] “A. M. Stëpin, A. M. Eremenko”, Math., 195:12 (2004), 1795–1808 | Zbl

[43] J.-P. Thouvenot, “On the stability of the weak Pinsker property”, Israel J. Math., 27:2 (1977), 150–162 | DOI | MR | Zbl

[44] S. V. Tikhonov, “Embeddings lattice actions in flows with multidimensional time”, Sb. Math., 197:1-2 (2006), 95–126 | DOI | MR | Zbl

[45] W. A. Veech, “Point-distal flows”, Amer. J. Math., 92 (1970), 205–242 | DOI | MR | Zbl

[46] W. A. Veech, “A criterion for a process to be prime”, Monatsh. Math., 94 (1982), 335–341 | DOI | MR | Zbl