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2000 | 84/85 | 2 | 521-547
Tytuł artykułu

The generic transformation has roots of all orders

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In the sense of the Baire Category Theorem we show that the generic transformation T has roots of all orders (RAO theorem). The argument appears novel in that it proceeds by establishing that the set of such T is not meager - and then appeals to a Zero-One Law (Lemma 2). On the group Ω of (invertible measure-preserving) transformations, §D shows that the squaring map p: S → S^{2} is topologically complex in that both the locally-dense and locally-lacunary points of p are dense (Theorem 23). The last section, §E, discusses the relation between RAO and a recent example of Blair Madore. Answering a question of the author's, Madore constructs a transformation with a square-root chain of each finite length, yet possessing no infinite square-root chain.
Słowa kluczowe
Rocznik
Tom
Numer
2
Strony
521-547
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-11-08
poprawiono
2000-02-20
Twórcy
  • University of Florida, PO Box 118105 Gainesville, FL 32611-2082, U.S.A.
Bibliografia
  • [1] F. Beleznay, The complexity of the collection of countable linear orders of the form (I+I), preprint, 1997.
  • [2] K. R. Berg, Mixing, cyclic approximation, and homomorphisms, Maryland Technical Report, 1975.
  • [3] N. Friedman, P. Gabriel, and J. L. King, An invariant for rigid rank-1 transformations, Ergodic Theory Dynam. Systems 8:53-72, 1988.
  • [4] E. Glasner and J. L. King, A zero-one law for dynamical properties, in: Topological Dynamics and Applications (Minneapolis, MN, 1995), Amer. Math. Soc., Providence, RI, 1998, 231-242.
  • [5] P. R. Halmos, Lectures on Ergodic Theory, Publ. Math. Soc. Japan 3, Math. Soc. Japan, 1956.
  • [6] P. D. Humke and M. Laczkovich, The Borel structure of iterates of continuous functions, Proc. Edinburgh Math. Soc. (2) 32:483-494, 1989.
  • [7] A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynam. Systems 7:531-557, 1987.
  • [8] A. B. Katok, Ya. G. Sinaĭ, and A. M. Stepin, The theory of dynamical systems and general transformation groups with invariant measure, (errata insert), Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Akad. Nauk SSSR, Moscow, 1975, 129-262 (in Russian).
  • [9] A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Uspekhi Mat. Nauk 22 (1967), no. 5, 81-106.
  • [10] J. L. King, The commutant is the weak closure of the powers, for rank-1 transformations, Ergodic Theory Dynam. Systems 6:363-384, 1986.
  • [11] J. L. King, Joining-rank and the structure of finite rank mixing transformations, J. Anal. Math. 51:182-227, 1988.
  • [12] K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966.
  • [13] B. F. Madore, Rank-one mixing actions with simple ℤ subactions, Doctoral dissertation, 2000.
  • [14] D. S. Ornstein, On the root problem in ergodic theory, in: Proc. Sixth Berkeley Sympos. Math. Statist. Probab. (Berkeley, CA, 1970/1971), Vol. II: Probability Theory, Univ. California Press, Berkeley, CA, 1972, 347-356.
  • [15] D. J. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. Anal. Math. 35:97-122, 1979
  • [16] W. A. Veech, A criterion for a process to be prime, Monatsh. Math. 94:335-341, 1982.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv84i2p521bwm
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