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Generic spectral properties of measure-preserving maps and applications

Authors: Andrés del Junco and Mariusz Lemańczyk
Journal: Proc. Amer. Math. Soc. 115 (1992), 725-736
MSC: Primary 28D05; Secondary 47A35
MathSciNet review: 1079889
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Abstract: Let $ \mathcal{K}$ denote the group of all automorphisms of a finite Lebesgue space equipped with the weak topology. For $ T \in \mathcal{K}$, let $ {\sigma _T}$ denote its maximal spectral type.

Theorem 1. There is a dense $ {G_\delta }$ subset $ G \subset \mathcal{K}$ such that, for each $ T \in G$ and $ k(1), \ldots ,k(l) \in {\mathbb{Z}^ + },k'(1), \ldots ,k'(l') \in {\mathbb{Z}^ + }$, the convolutions

$\displaystyle {\sigma _{{T^{k(1)}}}}* \cdots *{\sigma _{{T^{k(l)}}}}\quad and\quad {\sigma _{{T^{k'(1)}}}}* \cdots *{\sigma _{{T^{k'(l')}}}}$

are mutually singular, provided that ( $ (k(1), \ldots ,k(l))$) is not a rearrangement of $ (k'(1), \ldots ,k'(l'))$.

Theorem 1 has the following consequence.

Theorem 2. $ \mathcal{K}$ has a dense $ {G_\delta }$ subset $ F \subset G$ such that for $ T \in F$ the following holds: For any $ {\mathbf{k}}:\mathbb{N} \to \mathbb{Z} - \{ 0\} $ and $ l \in \mathbb{Z} - \{ 0\} $, the only way that $ {T^l}$, or any factor thereof, can sit as a factor in $ {T^{{\mathbf{k}}(1)}} \times {T^{{\mathbf{k}}(2)}} \times \cdots $ is inside the $ i$th coordinate $ \sigma $-algebra for some $ i$ with $ {\mathbf{k}}(i) = l$.

Theorem 2 has applications to the construction of certain counterexamples, in particular nondisjoint automorphisms having no common factors and weakly isomorphic automorphisms that are not isomorphic.

References [Enhancements On Off] (What's this?)

  • [ACaS] M. A. Akcoglu, R. V. Chacón, and T. Schwartzbauer, Commuting transformations and mixing, Proc. Amer. Math. Soc. 24 (1970), 637-642. MR 0254212 (40:7421)
  • [CoN] J. R. Choksi and M. G. Nadkarni, Baire category in spaces of measure, unitary operators and transformations, preprint.
  • [Fe] S. Ferenczi, Systemes localement de rang un, Ann. Inst. Henri Poincaré 20 (1984), 35-51. MR 740249 (85f:28013)
  • [Fu] H. Furstenburg, Disjointness in ergodic theory, minimal sets and a problem in diophantine approximation. Math. Systems Theory 1 (1967), 1-49. MR 0213508 (35:4369)
  • [HhP] F. Hahn and W. Parry, Some characteristic properties of dynamical systems with quasidiscrete spectrum, Math. Systems Theory 2 (1968), 179-190. MR 0230877 (37:6435)
  • [H] P. R. Halmos, Lectures on Ergodic Theory, Chelsea, New York, 1960. MR 0111817 (22:2677)
  • [J] A. del Junco, Disjointness of measure-preserving transformations, minimal self-joinings and category, Ergodic Theory Dynamical Systems I, Prog. Math., vol. 10, Birkhäuser, Boston, 1981, pp. 81-89. MR 633762 (82m:28035)
  • [JR] A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynamical Systems 7 (1987), 531-557. MR 922364 (89e:28029)
  • [K] A. B. Katok, Constructions in ergodic theory, preprint.
  • [L1] M. Lemańczyk, On the weak isomorphism of strictly ergodic homeomorphisms, Monatsh. Math. 108 (1989), 39-46. MR 1018823 (90g:28022)
  • [L2] M. Lemańczyk, Weakly isomorphic transformations that are not isomorphic, Probab. Theory Related Fields 78 (1988), 491-507. MR 950343 (89h:28028)
  • [LM] M. Lemańczyk and M. K. Mentzen, On metric properties of substitutions, Compositio Math. 65 (1988), 241-263. MR 932072 (89c:28023)
  • [N] D. Newton, Coalescence and spectrum of automorphisms of a Lebesgue space, Z. Wahrsch. Verw. Gebiete 19 (1971), 117-122. MR 0289748 (44:6936)
  • [O] D. S. Ornstein, On the root problem in ergodic theory, Proc. Sixth Berkeley Sympos. Math. Stat. Prob. Vol II, Univ. of California Press, 1967, pp. 347-356 MR 0399415 (53:3259)
  • [ORW] D. S. Ornstein, D. J. Rudolph and B. Weiss, Equivalence of measure-preserving transformations, Mem. Amer. Math. Soc., no. 262, Amer. Math. Soc., Providence, RI, 1982. MR 653094 (85e:28026)
  • [R] D. J. Rudolph, An example of a measure-preserving map with minimal self-joinings and applications, J. Anal. Math. 35 (1979), 97-122. MR 555301 (81e:28011)
  • [Si] Ya. G. Sinai, On weak isomorphism of transformations with invariant measure, Mat. Sb. 63 (1963), 23-42, (Russian)
  • [St] A. M. Stepin, Spectral properties of generic dynamical systems, Math. U.S.S.R. Izv. 29, 159-192.
  • [T] J.-P. Thouvenot, The metrical structure of some Gaussian processes, Proc. Ergodic Theory and Rel. Topics II, Georgenthal, 1986, pp. 195-198. MR 931147 (89m:28029)
  • [V] W. Veech, A criterion for a process to be prime, Monatsh. Math. 94 (1982), 335-341. MR 685378 (84d:28026)

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