×

Funny rank-one weak mixing for nonsingular Abelian actions. (English) Zbl 1024.28014

A nonsingular action \(S\) of a locally compact second countable group \(G\) on a \(\sigma\)-finite Lebesgue space \((Y,{\mathcal A},\nu)\) has funny rank one if there is a sequence \((Y_n)^\infty_{n=1}\) of measurable subsets of \(Y\) and a sequence \((G_n)^\infty_{n-1}\) of finite \(G\)-subsets such that
1. the sequences \(S_gY_n\), \(g\in G_n\), are pairwise disjoint for each \(n\),
2. given \(A\in{\mathcal A}\) of finite measure, then \(\inf_{P\subset G_n}\nu(A\Delta\bigcup_{g\in P} S_gY_n)\to 0\) as \(n\to\infty\),
3. \(\sum_{g\in G_n} \inf_{r\in R} \int_{Y_n} \left|{d\nu\circ S_g\over d\nu}- r\right|d\nu\to 0\) as \(n\to\infty\).
Two finite subsets \(C_1\) and \(C_2\) of \(G\) are called independent if \((C_1- C_1)\cap (C_2- C_2)= \{0\}\), and a sequence \((C_n)^\infty_{n=1}\) is independent if \(C_1+\cdots+ C_n\) and \(C_{n+1}\) is independent for all \(n\). Let \((C_n)^\infty_{n=1}\) and \((F_n)^\infty_{n=0}\) be two sequences of finite \(G\)-subset with \(F_0= \{0\}\), and for \(n> 0\)
1. \(F_n+ C_{n+1}\subset F_{n+1}\), \(\#(C_n)> 1\),
2. \(F_n,C_{n+1},C_{n+2},\dots\) is independent.
Put \(X_n= F_n\times \prod_{k> n} C_k\), and define a map \(i_n: X_n\to X_{n+1}\): \[ i_n(f_n, c_{n+1}, c_{n+2},\dots)= (f_n+ c_{n+1},c_{n+2},\dots). \] Denote by \(X\) the topological inductive limit of \((X_n,i_n)\) and by \(\widehat i_n: X_n\to X\) the canonical embeddings. Assume that for a given \(g\in G\), there exists \(m\) and \(g+ F_n+ C_{n+1}\subset F_{n+1}\) for \(n> m\). Set \[ D^{(n)}_g= (F_n\cap (F_n- g))\times \prod_{k> n} C_k,\quad\text{and }R^{(n)}_g= D^{(n)}_{-g}. \] Define \(T^{(n)}_g: D^{(n)}_g\to R_g(n)\) by \[ T^{(n)}_g(f_n, c_{n+1}, c_{n+2},\dots)= (f_n+ g, c_{n+1}, c_{n+2},\dots). \] Now put \(D_g= \bigcup^\infty_{n=1}\widehat i_n(D^{(n)}_g)\) and \(R_g= \bigcup^\infty_{n=1}\widehat i_n(R^{(n)}_g)\). Then there exists a homeomorphism \(T_g: D_g\to R_g\). \(T= \{T_g\}_{g\in G}\) is called \((C,F)\)-action. The main theorem in this article concerns the construction of various infinite measure preserving transformations.
Theorem 1. Let \(G\) be a countable Abelian group. For each one of the followings:
1. for every \(g\in G\) of infinite order, the transformation \(T_g\) has infinite ergodic index,
2. for each finite sequence \(g_1,\dots, g_n\) of \(G\)-elements of infinite order, the transformation \(T_{g_1}\times\cdots\times T_{g_n}\) is ergodic,
3. for each \(g\in G\) of infinite order, \(T_g\) has infinite ergodic index but \(T_{2g}\times T_g\) is nonconservative,
4. the Cartesian square of \(T\) is nonconservative,
5. \(T\) has trivial \(L^\infty\)-spectrum, nonergodic Cartesian square but all \(k\)-fold Cartesian products conservative,
there exists a funny rank one infinite measure preserving free \((C,F)\)-action \(T\) of \(G\).
A topological version of this theorem is also given. Moreover, given an \(AT\)-flow \(W\), a nonsingular \(G\)-action \(T\) with similar properties and such that the associated flow of \(T\) is \(W\) is constructed.
Reviewer: Makoto Mori

MSC:

28D15 General groups of measure-preserving transformations
37A25 Ergodicity, mixing, rates of mixing
37A15 General groups of measure-preserving transformations and dynamical systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, T.; Friedman, N.; Silva, C. E., Rank-one weak mixing for non-singular transformations, Israel Journal of Mathematics, 102, 269-281 (1997) · Zbl 0896.58039
[2] [AFS2] T. Adams, N. Friedman and C. E. Silva,Rank one power weak mixing non-singular transformations, Ergodic Theory and Dynamical Systems, to appear.
[3] Aaronson, J.; Lin, M.; Weiss, B., Mixing properties of Markov operators and ergodic transformations, and ergodicity of Cartesian products, Israel Journal of Mathematics, 33, 198-224 (1979) · Zbl 0438.28018 · doi:10.1007/BF02762161
[4] Connes, A.; Woods, E. J., Approximately transitive flows and ITPFI factors, Ergodic Theory and Dynamical Systems, 5, 203-236 (1985) · Zbl 0606.46041
[5] [Da] A. I. Danilenko,Strong orbit equivalence of locally compact Cantor minimal systems, International Journal of Mathematics, to appear. · Zbl 1110.37300
[6] Day, S. L.; Grivna, B. R.; MaCartney, E. P.; Silva, C. E., Power weakly mixing infinite transformations, New York Journal of Mathematics, 5, 17-24 (1999) · Zbl 0923.28006
[7] Eigen, S.; Haijan, A.; Ito, Y., Ergodic measure preserving transformations of finite type, Tokyo Journal of Mathematics, 11, 459-470 (1988) · Zbl 0675.28008 · doi:10.3836/tjm/1270133989
[8] Ferenczi, S., Systèmes de rang un gauche, Annales de l’Institut Henri Poincaré. Probabilités et Statistiques, 21, 177-186 (1985) · Zbl 0575.28013
[9] Feldman, J.; Moore, C. C., Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Transactions of the American Mathematical Society, 234, 289-324 (1977) · Zbl 0369.22009 · doi:10.2307/1997924
[10] Friedman, N. A., Introduction to Ergodic Theory (1970), New York: Van Nostrand, New York · Zbl 0212.40004
[11] Hamachi, T., A measure theoretical proof of the Connes-Woods theorem on AT-flows, Pacific Journal of Mathematics, 154, 67-85 (1992) · Zbl 0792.46048
[12] [HO] T. Hamachi and M. Osikawa,Ergodic groups of automorphisms and Krieger’s theorems, Seminar on Mathematical Sciences, Vol. 3, Keio University, 1981. · Zbl 0472.28015
[13] Hawkins, J. M., Properties of ergodic flows associated with product odometers, Pacific Journal of Mathematics, 141, 287-297 (1990) · Zbl 0734.28018
[14] Hajian, A.; Kakutani, S., An example of an ergodic measure preserving transformation defined on an infinite measure space, 45-52 (1970), Berlin: Springer-Verlag, Berlin · Zbl 0213.07601
[15] del Junco, A., A simple map with no prime factors, Israel Journal of Mathematics, 104, 301-320 (1998) · Zbl 0915.28011 · doi:10.1007/BF02897068
[16] Kakutani, S.; Parry, W., Infinite measure preserving transformations with “mixing”, Bulletin of the American Mathematical Society, 69, 752-756 (1963) · Zbl 0126.31801
[17] Muehlegger, E. J.; Raich, A. S.; Silva, C. E.; Touloumtzis, M. P.; Narasimhan, B.; Zhao, W., Infinite ergodic index ℤ^d in infinite measure, Colloquium Mathematicum, 82, 167-190 (1999) · Zbl 0940.28014
[18] [Sc] K. Schmidt,Lectures on cocycles of ergodic transformation groups, Lecture Notes in Mathematics, Vol. 1, Macmillan Co. of India, 1977. · Zbl 0421.28017
[19] [So] A. Sokhet,Les actions approximativement transitives dans la théory ergodique, Thèse de doctorat, Université Paris VII, 1997.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.