Isometric extensions, 2-cocycles and ergodicity of skew products

• Autorzy: Alexandre I. Danilenko , Mariusz Lemańczyk
• Języki publikacji: EN

Abstrakty

EN

We establish existence and uniqueness of a canonical form for isometric extensions of an ergodic non-singular transformation T. This is applied to describe the structure of commutors of the isometric extensions. Moreover, for a compact group G, we construct a G-valued T-cocycle α which generates the ergodic skew product extension $T_α$ and admits a prescribed subgroup in the centralizer of $T_α$.

Słowa kluczowe

EN

ergodic transformation; cocycle; isometric extension

Kategorie tematyczne

• 28D05: Measure-preserving transformations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1999
• Tom: 137
• Numer: 2
• Strony: 123-142

Daty

wydano

1999

otrzymano

1997-09-22

poprawiono

1999-08-30

Twórcy

autor

Alexandre I. Danilenko

• Department of Mechanics and Mathematics, Kharkov State University, Freedom sq. 4 Kharkov, 310077, Ukraine

autor

Mariusz Lemańczyk

• Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Bibliografia

• [Br] L. G. Brown, Topologically complete groups, Proc. Amer. Math. Soc. 35 (1972), 593-600.
• [D1] A. I. Danilenko, Comparison of cocycles of measured equivalence relations and lifting problems, Ergodic Theory Dynam. Systems 18 (1998), 125-151.
• [D2] A. I. Danilenko, On cocycles with values in group extensions. Generic results, Mat. Analiz Geom., to appear.
• [DG] A. I. Danilenko and V. Ya. Golodets, Extension of cocycles to normalizer elements, outer conjugacy and related problems, Trans. Amer. Math. Soc. 348 (1996), 4857-4882.
• [FL] S. Ferenczi and M. Lemańczyk, Rank is not a spectral invariant, Studia Math. 98 (1991), 227-230.
• [GLS] P. Gabriel, M. Lemańczyk and K. Schmidt, Extensions of cocycles for hyperfinite actions and applications, Monatsh. Math. 123 (1997), 209-228.
• [Ha] T. Hamachi, On a minimal group cover of an ergodic finite extension, preprint.
• [JLM] A. del Junco, M. Lemańczyk and M. Mentzen, Semisimplicity, joinings, and group extensions, Studia Math. 112 (1995), 141-164.
• [Ki] J. King, The commutant is the weak closure of the powers, for rank-1 transformations, Ergodic Theory Dynam. Systems 6 (1986), 363-384.
• [Kw] J. Kwiatkowski, Factors of ergodic group extensions of rotations, Studia Math. 103 (1992), 123-131.
• [Le] M. Lemańczyk, Cohomology groups, multipliers and factors in ergodic theory, ibid. 122 (1997), 275-288.
• [Me] M. K. Mentzen, Ergodic properties of group extensions of dynamical systems with discrete spectra, ibid. 101 (1991), 20-31.
• [Ne] D. Newton, On canonical factors of ergodic dynamical systems, J. London Math. Soc. 19 (1979), 129-136.
• [Pa] K. R. Parthasarathy, Multipliers on Locally Compact Groups, Lecture Notes in Math. 93, Springer, 1969.
• [Sc] K. Schmidt, Lectures on Cocycles of Ergodic Transformation Groups, Lecture Notes in Math. 1, Macmillan, 1977.
• [Z1] R. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), 373-409.
• [Z2] R. Zimmer, Ergodic Theory and Semisimple Lie Groups, Birkhäuser, Boston, 1984.