On asymmetry of the future and the past for limit self-joinings. (English) Zbl 1024.37003
Author’s abstract: Let \(\Delta_T\) be an off-diagonal joining of a transformation \(T\). We construct a nontypical transformation having asymmetry between limit sets of \(\Delta_{T^n}\) for positive and negative powers of \(T\). It follows from a correspondence between subpolymorphisms and positive operators, and from the structure of limit polynomial operators. We apply this technique to find all polynomial operators of degree 1 in the weak closure (in the space of positive operators on \(L_2\)) of powers of Chacon’s automorphism and its generalizations.
Reviewer: Lasha Ephremidze (Tiblisi)
MSC:
| 37A05 | Dynamical aspects of measure-preserving transformations |
| 47B65 | Positive linear operators and order-bounded operators |
| 28D05 | Measure-preserving transformations |
| 47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |
Keywords:
joinings; Chacon’s automorphism; weak operator convergence; subpolymorphisms; positive operators; polynomial operatorsReferences:
| [1] | Daniel J. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. Analyse Math. 35 (1979), 97 – 122. · Zbl 0446.28018 · doi:10.1007/BF02791063 |
| [2] | Marina Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2) 118 (1983), no. 2, 277 – 313. · Zbl 0556.28020 · doi:10.2307/2007030 |
| [3] | A. del Junco, M. Rahe, and L. Swanson, Chacon’s automorphism has minimal self-joinings, J. Analyse Math. 37 (1980), 276 – 284. · Zbl 0445.28014 · doi:10.1007/BF02797688 |
| [4] | Jonathan King, The commutant is the weak closure of the powers, for rank-1 transformations, Ergodic Theory Dynam. Systems 6 (1986), no. 3, 363 – 384. · Zbl 0595.47005 · doi:10.1017/S0143385700003552 |
| [5] | Bernard Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Israel J. Math. 76 (1991), no. 3, 289 – 298. · Zbl 0790.28010 · doi:10.1007/BF02773866 |
| [6] | A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynam. Systems 7 (1987), no. 4, 531 – 557. · Zbl 0646.60010 · doi:10.1017/S0143385700004193 |
| [7] | A. M. Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples, Uspekhi Mat. Nauk 55 (2000), no. 4(334), 59 – 128 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 4, 667 – 733. · Zbl 0991.37005 · doi:10.1070/rm2000v055n04ABEH000314 |
| [8] | O. N. Ageev, On ergodic transformations with homogeneous spectrum, J. Dynam. Control Systems 5 (1999), no. 1, 149 – 152. · Zbl 0943.37005 · doi:10.1023/A:1021701019156 |
| [9] | V. V. Ryzhikov, Transformations having homogeneous spectra, J. Dynam. Control Systems 5 (1999), no. 1, 145 – 148. · Zbl 0954.37007 · doi:10.1023/A:1021748902318 |
| [10] | A. B. Katok, Ja. G. Sinaĭ, and A. M. Stepin, The theory of dynamical systems and general transformation groups with invariant measure, Mathematical analysis, Vol. 13 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1975, pp. 129 – 262. (errata insert) (Russian). |
| [11] | G. R. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems 5 (1999), no. 2, 173 – 226. · Zbl 0987.37004 · doi:10.1023/A:1021726902801 |
| [12] | E. A. Robinson Jr., Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), no. 2, 299 – 314. · Zbl 0519.28008 · doi:10.1007/BF01389325 |
| [13] | O. N. Ageev, The spectral multiplicity function and geometric representations of interval exchange transformations, Mat. Sb. 190 (1999), no. 1, 3 – 28 (Russian, with Russian summary); English transl., Sb. Math. 190 (1999), no. 1-2, 1 – 28. · Zbl 0934.28009 · doi:10.1070/SM1999v190n01ABEH000376 |
| [14] | Daniel J. Rudolph, Fundamentals of measurable dynamics, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990. Ergodic theory on Lebesgue spaces. · Zbl 0718.28008 |
| [15] | O. N. Ageev, On the spectrum of Cartesian powers of classical automorphisms, Mat. Zametki 68 (2000), no. 5, 643 – 647 (Russian, with Russian summary); English transl., Math. Notes 68 (2000), no. 5-6, 547 – 551. · Zbl 1042.37003 · doi:10.1023/A:1026698921311 |
| [16] | O. Ageev, C. Silva, Genericity of rigidity and multiple recurrence for infinite measure preserving and nonsingular transformations, preprint. · Zbl 1082.37502 |
| [17] | M. Lemańczyk, F. Parreau, and J.-P. Thouvenot, Gaussian automorphisms whose ergodic self-joinings are Gaussian, Fund. Math. 164 (2000), no. 3, 253 – 293. · Zbl 0977.37003 |
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.
