On the spectral type of Ornstein’s class one transformations.

*(English)*Zbl 0787.28011The rank one transformations are, roughly speaking, those measure- preserving transformations which may be obtained from the cutting and stacking of a bounded interval using exactly one column. Many classical examples are well-known, such as those of von Neumann, Kakutani and Chacon. Rank one transformations are always ergodic and it was shown by Baxter that they have simple spectrum. Using probabilistic methods, D. S. Ornstein [Proc. 6 Berkeley Sympos., Math. Stat. Prob., Univ. Calif. Vol. II, 1970, 347-356 (1972; Zbl 0262.28009)] constructed a class of such transformations which are mixing. There has been speculation that the rank one mixing transformations may have non-singular spectrum, or in fact simple Lebesgue spectrum (which would answer a question due to Banach). However, the author shows that any member of Ornstein’s class of rank one mixing transformations has singular spectrum (i.e., its maximal spectral type is singular with respect to Lebesgue measure). The author discusses the question of whether all rank one transformations have singular spectrum, a question which is still open. The paper, which is quite technical, uses harmonic analysis techniques.

Reviewer: G.R.Goodson (Towson)

##### Keywords:

spectral multiplicity; rank one transformations; measure-preserving transformations; cutting; stacking; rank one mixing transformations; Ornstein’s class; singular spectrum; maximal spectral type
Full Text:
DOI

##### References:

[1] | A. Bonami,Ensembles \(\Lambda\)(p) dans le dual de D nn. Inst. Fourier (Grenoble)18 (1968), 293–304. |

[2] | B. Host,Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Israel J. Math.76 (1991), 289–298. · Zbl 0790.28010 · doi:10.1007/BF02773866 |

[3] | S. Kalikow,Twofold mixing implies threefold mixing for rank one transformations, Ergodic Theory and Dynamical Systems4 (1984), 237–259. · Zbl 0552.28016 · doi:10.1017/S014338570000242X |

[4] | C. McGehee, O.C. Pigno and B. Smith,Hardy’s inequality and the L 1-norm of exponential sums, Annals of Math.113 (1987), 613–618. · Zbl 0473.42001 · doi:10.2307/2007000 |

[5] | D. Ornstein,On the root problem in ergodic theory, Proc. Sixth Berkeley Symp. Math. Stat. and Prob., Vol. II, pp. 347–356. |

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.