Cut-and-stack simple weakly mixing map with countably many prime factors

Authors:
Alexandre I. Danilenko and Andres del Junco

Journal:
Proc. Amer. Math. Soc. **136** (2008), 2463-2472

MSC (2000):
Primary 37A05

Published electronically:
March 4, 2008

MathSciNet review:
2390514

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Abstract | References | Similar Articles | Additional Information

Abstract: Via the cut-and-stack construction we produce a 2-fold simple weakly mixing transformation which has countably many proper factors and all of them are 2-to-1 and prime.

**[Da1]**Alexandre I. Danilenko,*Funny rank-one weak mixing for nonsingular abelian actions*, Israel J. Math.**121**(2001), 29–54. MR**1818379**, 10.1007/BF02802494**[Da2]**Alexandre I. Danilenko,*Mixing rank-one actions for infinite sums of finite groups*, Israel J. Math.**156**(2006), 341–358. MR**2282382**, 10.1007/BF02773838**[Da3]**Alexandre I. Danilenko,*Explicit solution of Rokhlin’s problem on homogeneous spectrum and applications*, Ergodic Theory Dynam. Systems**26**(2006), no. 5, 1467–1490. MR**2266369**, 10.1017/S0143385706000344**[Da4]**Alexandre I. Danilenko,*On simplicity concepts for ergodic actions*, J. Anal. Math.**102**(2007), 77–117. MR**2346554**, 10.1007/s11854-007-0017-x**[Da5]**-,*Weakly mixing rank-one actions conjugate to their squares*, Preprint.**[DaS]**Alexandre I. Danilenko and Cesar E. Silva,*Mixing rank-one actions of locally compact abelian groups*, Ann. Inst. H. Poincaré Probab. Statist.**43**(2007), no. 4, 375–398 (English, with English and French summaries). MR**2329508**, 10.1016/j.anihpb.2006.05.002**[dJ]**Andrés del Junco,*A simple map with no prime factors*, Israel J. Math.**104**(1998), 301–320. MR**1622315**, 10.1007/BF02897068**[dJR]**A. del Junco and D. Rudolph,*On ergodic actions whose self-joinings are graphs*, Ergodic Theory Dynam. Systems**7**(1987), no. 4, 531–557. MR**922364**, 10.1017/S0143385700004193**[Fe]**S. Ferenczi,*Systèmes de rang un gauche*, Ann. Inst. H. Poincaré Probab. Statist.**21**(1985), no. 2, 177–186 (French, with English summary). MR**798894****[GHR]**E. Glasner, B. Host, and D. Rudolph,*Simple systems and their higher order self-joinings*, Israel J. Math.**78**(1992), no. 1, 131–142. MR**1194963**, 10.1007/BF02801575**[GlW]**Eli Glasner and Benjamin Weiss,*A simple weakly mixing transformation with nonunique prime factors*, Amer. J. Math.**116**(1994), no. 2, 361–375. MR**1269608**, 10.2307/2374933**[Li]**Elon Lindenstrauss,*Pointwise theorems for amenable groups*, Invent. Math.**146**(2001), no. 2, 259–295. MR**1865397**, 10.1007/s002220100162**[Ma]**Blair Madore,*Rank-one group actions with simple mixing ℤ-subactions*, New York J. Math.**10**(2004), 175–194 (electronic). MR**2114785****[Or]**Donald S. Ornstein,*On the root problem in ergodic theory*, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 347–356. MR**0399415****[Ra]**Marina Ratner,*Horocycle flows, joinings and rigidity of products*, Ann. of Math. (2)**118**(1983), no. 2, 277–313. MR**717825**, 10.2307/2007030**[So]**A. Sokhet,*Les actions approximativement transitives dans la théory ergodique*, Thèse de doctorat, Université Paris VII, 1997.**[Th]**J.-P. Thouvenot,*Some properties and applications of joinings in ergodic theory*, Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), London Math. Soc. Lecture Note Ser., vol. 205, Cambridge Univ. Press, Cambridge, 1995, pp. 207–235. MR**1325699**, 10.1017/CBO9780511574818.004**[Ve]**William A. Veech,*A criterion for a process to be prime*, Monatsh. Math.**94**(1982), no. 4, 335–341. MR**685378**, 10.1007/BF01667386

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Additional Information

**Alexandre I. Danilenko**

Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, Bonn, 53111, Germany

Address at time of publication:
Institute for Low Temperature Physics & Engineering of National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov, 61164, Ukraine

Email:
danilenko@ilt.kharkov.ua

**Andres del Junco**

Affiliation:
Department of Mathematics, University of Toronto, Toronto, M5S 3G3, Canada

Email:
deljunco@math.toronto.edu

DOI:
https://doi.org/10.1090/S0002-9939-08-09154-5

Keywords:
Joining,
2-fold simple transformation,
$(C,F)$-construction

Received by editor(s):
November 27, 2006

Received by editor(s) in revised form:
February 22, 2007

Published electronically:
March 4, 2008

Communicated by:
Jane M. Hawkins

Article copyright:
© Copyright 2008
American Mathematical Society