On the Foiaş and Strătilă theorem. (English) Zbl 07887999
Summary: We extend the Foiaş and Strătilă theorem to the case of \(L^{2}\)-functions whose spectral measures are continuous and concentrated on an independent Helson set, and to ergodic actions of locally compact second countable abelian groups. We first prove it for functions satisfying Carleman’s condition for the Hamburger moment problem, without the assumption that the spectral measure is supported by a Helson set. Then we show independently that the spectral projector associated with a Helson set preserves each \(L^{p}\)-space, with an appropriate bound of the corresponding norm.
MSC:
| 37A46 | Relations between ergodic theory and harmonic analysis |
| 37A15 | General groups of measure-preserving transformations and dynamical systems |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 43A46 | Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) |
Keywords:
ergodic action of groups; spectral measure; Gaussian process; Foiaş and Strătilă; Helson set; Carleman’s conditionReferences:
| [1] | O. N. Ageev, On the spectrum of cartesian powers of classical automorphisms, Math. Notes 68 (2000), 547-551. · Zbl 1042.37003 |
| [2] | I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai, Ergodic Theory, Springer, New York, 1982. · Zbl 0493.28007 |
| [3] | A. I. Danilenko and V. V. Ryzhikov, Mixing constructions with infinite invariant measure and spectral multiplicities, Ergodic Theory Dynam. Systems 31 (2011), 853-873. · Zbl 1246.37010 |
| [4] | Y. Derriennic, K. Frączek, M. Lemańczyk, and F. Parreau, Ergodic automorphisms whose weak closure of off-diagonal measures consists of ergodic self-joinings, Colloq. Math. 110 (2008), 81-115. · Zbl 1142.37005 |
| [5] | J. L. Doob, Stochastic Processes, Wiley, New York, 1953. · Zbl 0053.26802 |
| [6] | S. W. Drury, Sur les ensembles de Sidon, C. R. Acad. Sci. Paris 271 (1970), 162-163. · Zbl 0194.42804 |
| [7] | C. Foiaş, Sur les mesures spectrales qui interviennent dans la théorie ergodique, J. Math. Mech. 13 (1964), 639-658. · Zbl 0163.37501 |
| [8] | C. Foiaş, Projective tensor products in ergodic theory, Rev. Roumaine Math. Pures Appl. 12 (1967), 1221-1226. · Zbl 0164.16001 |
| [9] | C. Foiaş et S. Strătilă, Ensembles de Kronecker dans la théorie ergodique, C. R. Acad. Sci. Paris 267 (1968), 166-168. · Zbl 0218.60040 |
| [10] | C. S. Herz, Drury’s lemma and Helson sets, Studia Math. 42 (1972), 205-219. · Zbl 0229.43009 |
| [11] | T. W. Körner, Some results on Kronecker, Dirichlet and Helson sets, Ann. Inst. Fourier (Grenoble) 20 (1970), 219-324. · Zbl 0196.08403 |
| [12] | J. Kułaga-Przymus and F. Parreau, Disjointness properties for cartesian products of weakly mixing systems, Colloq. Math. 128 (2011), 153-177. · Zbl 1268.37005 |
| [13] | P. Lefèvre and L. Rodríguez-Piazza, p-Rider sets are q-Sidon sets, Proc. Amer. Math. Soc. 131 (2003), 1829-1838. · Zbl 1010.43006 |
| [14] | M. Lemańczyk and F. Parreau, On the disjointness problem for Gaussian automor-phisms, Proc. Amer. Math. Soc. 127 (1999), 2073-2081. · Zbl 0923.28007 |
| [15] | M. Lemańczyk, F. Parreau, and J.-P. Thouvenot, Gaussian automorphisms whose ergodic self-joinings are Gaussian, Fund. Math. 164 (2000), 253-293. · Zbl 0977.37003 |
| [16] | L. A. Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis, Dekker, New York, 1971. · Zbl 0226.43006 |
| [17] | J.-F. Méla, Mesures ε-idempotentes de norme bornée, Studia Math. 72 (1982), 131-149. · Zbl 0503.43004 |
| [18] | W. Parry, Topics in Ergodic Theory, Cambridge Tracts in Math. 75, Cambridge Univ. Press, Cambridge, 1981. · Zbl 0449.28016 |
| [19] | M. Queffélec, Substitution Dynamical Systems -Spectral Analysis, 2nd ed., Lecture Notes in Math. 1294, Springer, Berlin, 2010. · Zbl 1225.11001 |
| [20] | E. Roy, Poisson suspensions and infinite ergodic theory, Ergodic Theory Dynam. Systems 29 (2009), 667-683. · Zbl 1160.37303 |
| [21] | W. Rudin, Fourier Analysis on Groups, Wiley, New York, 1962. · Zbl 0107.09603 |
| [22] | W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966. · Zbl 0142.01701 |
| [23] | J.-P. Thouvenot, The metrical structure of some Gaussian processes, in: Proceedings of the Conference on Ergodic Theory and Related Topics, II (Georgenthal, 1986), Teubner, Leipzig, 1987, 195-198. · Zbl 0653.28014 |
| [24] | N. T. Varopoulos, Groups of continuous functions in harmonic analysis, Acta Math. 125 (1970), 109-154. · Zbl 0214.38102 |
| [25] | N. T. Varopoulos, Sur la réunion de deux ensembles de Helson, C. R. Acad. Sci. Paris 271 (1970), 251-253. · Zbl 0205.13103 |
| [26] | L. Zsido, Sur les processus fortement stationnaires ergodiques, C. R. Acad. Sci. Paris 267 (1968), 169-170. · Zbl 0159.46203 |
| [27] | François Parreau Université Sorbonne Paris Nord LAGA, CNRS, UMR 7539 |
| [28] | F-93430 Villetaneuse, France E-mail: parreau@math.univ-paris13.fr |
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.
