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## Colloquium Mathematicum

1999 | 82 | 2 | 167-190
Tytuł artykułu

### Infinite ergodic index $ℤ^d$ -actions in infinite measure

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We construct infinite measure preserving and nonsingular rank one $ℤ^d$-actions. The first example is ergodic infinite measure preserving but with nonergodic, infinite conservative index, basis transformations; in this case we exhibit sets of increasing finite and infinite measure which are properly exhaustive and weakly wandering. The next examples are staircase rank one infinite measure preserving $ℤ^d$-actions; for these we show that the individual basis transformations have conservative ergodic Cartesian products of all orders, hence infinite ergodic index. We generalize this example to obtain a stronger condition called power weakly mixing. The last examples are nonsingular $ℤ^d$-actions for each Krieger ratio set type with individual basis transformations with similar properties.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
167-190
Opis fizyczny
Daty
wydano
1999
otrzymano
1999-03-15
Twórcy
autor
• Lexecon Inc., One Mifflin Place, Cambridge, MA 02138, U.S.A.
autor
• 480 Lincoln Drive, Department of Mathematics, University of Wisconsin, Madison, Madison, WI 53706, U.S.A.
autor
• Department of Mathematics, Williams College, Williamstown, MA 01267, U.S.A.
autor
• 1750 41st Avenue, San Francisco, CA 94122, U.S.A.
autor
• Department of Mathematics, Williams College, Williamstown, MA 01267, U.S.A.
autor
• Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY 14853, U.S.A.
Bibliografia
• [ALW] J. Aaronson, M. Lin and B. Weiss, Mixing properties of Markov operators and ergodic transformations, and ergodicity of cartesian products, Israel J. Math. 33 (1979), 198-224.
• [A] T. Adams, Uniformly sweeping out, PhD thesis, State University of New York at Albany, 1991.
• [AFS] T. Adams, N. Friedman and C. Silva, Rank-one weak mixing for nonsingular transformations, Israel J. Math. 102 (1997), 269-281.
• [AFS2] T. Adams, N. Friedman and C. Silva, Rank one power weakly mixing for nonsingular transformations, preprint.
• [AS] T. Adams and C. Silva, $ℤ^d$-staircase actions, Ergodic Theory Dynam. Systems 19 (1999), 837-850.
• [C] J. Crabtree, Weakly wandering sets, B.A. thesis, Williams College, 1993.
• [DGMS] S. Day, B. Grivna, E. McCartney and C. Silva, Power weakly mixing infinite transformations, New York J. Math., to appear.
• [EHK] S. Eigen, A. Hajian and S. Kakutani, Complementing sets of integers-a result from ergodic theory, Japan. J. Math. 18 (1992), 205-211.
• [F] N. Friedman, Replication and stacking in ergodic theory, Amer. Math. Monthly 99 (1992), 31-41.
• [HI] A. Hajian and Y. Ito, Weakly wandering sets and invariant measures for a group of transformations, J. Math. Mech. 18 (1969), 1203-1216.
• [HK1] A. Hajian and S. Kakutani, Weakly wandering sets and invariant measures, Trans. Amer. Math. Soc. 110 (1964), 136-151.
• [HK2] A. Hajian and S. Kakutani, An example of an ergodic measure preserving transformation on an infinite measure space, in: Lecture Notes in Math. 160, Springer, 1970, 45-52.
• [HO] T. Hamachi and Osikawa, Ergodic groups of automorphisms and Krieger's theorems, Seminar on Math. Sci. 3, Keio Univ., 1981.
• [JK] L. K. Jones and U. Krengel, On transformations without finite invariant measure, Adv. Math. 12 (1974), 275-276.
• [JS] A. del Junco and C. Silva, Prime type $III_λ$ automorphisms: An instance of coding techniques applied to nonsingular maps, in: Algorithms, Fractals and Dynamics (Okayama/Kyoto, 1992), Y. Takahashi (ed.), Plenum Press, New York, 1995, 101-115.
• [KP] S. Kakutani and W. Parry, Infinite measure preserving transformations withi 'mixing', Bull. Amer. Math. Soc. 69 (1963), 752-756.
• [Kre] U. Krengel, Ergodic Theorems, de Gruyter Stud. Math. 6, de Gruyter, Berlin, 1985.
• [Kri] U. Krengel, On the Araki-Woods asymptotic ratio set and nonsingular transformations of a measure space, in: Lecture Notes in Math. 160, Springer, 1970, 158-177.
• [PR] K. Park and E. A. Robinson, Jr., The joinings within a class of $ℤ^2$ actions, J. Anal. Math. 57 (1991), 1-36.
• [Sch] K. Schmidt, Dynamical Systems of Algebraic Origin, Birkhäuser, 1995.
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Bibliografia
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