2021, 17: 213-265. doi: 10.3934/jmd.2021007

A prime system with many self-joinings

1. 

Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA

2. 

Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA

Received  May 14, 2019 Revised  November 15, 2020 Published  April 2021

Fund Project: JC: Partially supported by NSF grants DMS-135500 and DMS-452762, the Sloan foundation, Poincaré chair, and Warnock chair.
BK: Partially supported by NSF grant DMS-1800544.

We construct a rigid, rank 1, prime transformation that is not quasi-simple and whose self-joinings form a Poulsen simplex. This seems to be the first example of a prime system whose self-joinings form a Poulsen simplex.

Citation: Jon Chaika, Bryna Kra. A prime system with many self-joinings. Journal of Modern Dynamics, 2021, 17: 213-265. doi: 10.3934/jmd.2021007
References:
[1]

O. N. Ageev, The generic automorphism of a Lebesgue space conjugate to a $G$-extension for any finite abelian group $G$, Dokl. Akad. Nauk., 374 (2000), 439-442. 

[2]

O. N. Ageev, A typical dynamical system is not simple or semisimple, Ergodic Theory Dynam. Systems, 23 (2003), 1625-1636.  doi: 10.1017/S0143385703000075.

[3]

J. Chaika, Self joinings of rigid rank one transformations arise as strong operator topology limits of convex combinations of powers, arXiv: 1901.08695.

[4]

J. Chaika and A. Eskin, Self-joinings for $3$-IETs, to appear, J. Eur. Math. Soc. (JEMS).

[5]

A. I. Danilenko, On simplicity concepts for ergodic actions, J. Anal. Math., 102 (2007), 77-117.  doi: 10.1007/s11854-007-0017-x.

[6]

A. I. Danilenko and A. del Junco, Cut-and-stack simple weakly mixing map with countably many prime factors, Proc. Amer. Math. Soc., 136 (2008), 2463-2472.  doi: 10.1090/S0002-9939-08-09154-5.

[7]

A. del Junco, A simple map with no prime factors, Israel J. Math., 104 (1998), 301-320.  doi: 10.1007/BF02897068.

[8]

A. del Junco and D. J. Rudolph, A rank-one, rigid, simple, prime map, Ergodic Theory Dynam. Systems, 7 (1987), 229-247.  doi: 10.1017/S0143385700003977.

[9]

A. del JuncoM. Rahe and L. Swanson, Chacon's automorphism has minimal self joinings, J. Analyse Math., 37 (1980), 276-284.  doi: 10.1007/BF02797688.

[10]

S. FerencziC. Holton and L. Q. Zamboni, Joinings of three-interval exchange transformations, Ergodic Theory Dynam. Systems, 25 (2005), 483-502.  doi: 10.1017/S0143385704000811.

[11]

H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.  doi: 10.2307/2373137.

[12] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981. 
[13]

E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.

[14]

E. GlasnerB. Host and D. Rudolph, Simple systems and their higher order self-joinings, Israel J. Math., 78 (1992), 131-142.  doi: 10.1007/BF02801575.

[15]

E. Glasner and B. Weiss, A simple weakly mixing transformation with nonunique prime factors, Amer. J. Math., 116 (1994), 361-375.  doi: 10.2307/2374933.

[16]

P. R. Halmos, In general a measure preserving transformation is mixing, Ann. of Math. (2), 45 (1944), 786-792.  doi: 10.2307/1969304.

[17]

É. JanvresseE. Roy and T. de la Rue, Poisson suspensions and Sushis, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 1301-1334.  doi: 10.24033/asens.2346.

[18]

É. JanvresseA. A. Prikhod'koT. de la Rue and V. V. Ryzhikov, Weak limits of powers of Chacon's automorphism, Ergodic Theory Dynam. Systems, 35 (2015), 128-141.  doi: 10.1017/etds.2013.49.

[19]

É. Janvresse, T. de la Rue and V. Ryzhikov, Around King's rank-one theorems: Flows and $\Bbb Z^n$-actions, Dynamical Systems and Group Actions, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012,143–161. doi: 10.1090/conm/567/11233.

[20]

A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, in Handbook in Dynamical Systems, 1B, Elsevier, 2005,649–743. doi: 10.1016/S1874-575X(06)80036-6.

[21]

J. King, The commutant is the weak closure of the powers, for rank-$1$ transformations, Ergodic Theory Dynam. Systems, 6 (1986), 363-384.  doi: 10.1017/S0143385700003552.

[22]

J. L. King, Joining-rank and the structure of finite rank mixing transformations, J. Analyse Math., 51 (1988), 182-227.  doi: 10.1007/BF02791123.

[23]

J. King, Flat stacks, joining closure and genericity, preprint, available from: http://squash.1gainesville.com/PDF/flatstacks.pdf

[24]

M. LemańczykF. Parreau and E. Roy, Joining primeness and disjointness from infinitely divisible systems, Proc. Amer. Math. Soc., 139 (2011), 185-199.  doi: 10.1090/S0002-9939-2010-10457-4.

[25]

J. LindenstraussG. Olsen and Y. Sternfeld, The Poulsen simplex, Ann. Inst. Fourier (Grenoble), 28 (1978), 91-114.  doi: 10.5802/aif.682.

[26]

W. Parry, Zero entropy of distal and related transformations, in Topological Dynamics (Symp. at Colorado State University, (1967)), Benjamin, New York, 1968,383–389.

[27]

F. Parreau and E. Roy, Prime Poisson suspensions, Ergodic Theory Dynam. Systems, 35 (2015), 2216-2230.  doi: 10.1017/etds.2014.32.

[28]

D. J. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. Analyse Math., 35 (1979), 97-122.  doi: 10.1007/BF02791063.

[29]

V. V. Ryzhikov, Bounded ergodic constructions, disjointness, and weak limits of powers, Trans. Moscow Math. Soc., (2013), 165–171. doi: 10.1090/s0077-1554-2014-00214-4.

[30]

J.-P. Thouvenot, Les systèmes simples sont disjoints de ceux qui sont infiniment divisibles et plongeables dans un flot, Colloq. Math., 84/85 (2000), 481-483.  doi: 10.4064/cm-84/85-2-481-483.

[31]

W. A. Veech, A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341.  doi: 10.1007/BF01667386.

[32]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[33]

R. J. Zimmer, Extensions of ergodic actions and generalized discrete spectrum, Bull. Amer. Math. Soc., 81 (1975), 633-636.  doi: 10.1090/S0002-9904-1975-13770-0.

[34]

R. J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math., 20 (1976), 555-588.  doi: 10.1215/ijm/1256049648.

show all references

References:
[1]

O. N. Ageev, The generic automorphism of a Lebesgue space conjugate to a $G$-extension for any finite abelian group $G$, Dokl. Akad. Nauk., 374 (2000), 439-442. 

[2]

O. N. Ageev, A typical dynamical system is not simple or semisimple, Ergodic Theory Dynam. Systems, 23 (2003), 1625-1636.  doi: 10.1017/S0143385703000075.

[3]

J. Chaika, Self joinings of rigid rank one transformations arise as strong operator topology limits of convex combinations of powers, arXiv: 1901.08695.

[4]

J. Chaika and A. Eskin, Self-joinings for $3$-IETs, to appear, J. Eur. Math. Soc. (JEMS).

[5]

A. I. Danilenko, On simplicity concepts for ergodic actions, J. Anal. Math., 102 (2007), 77-117.  doi: 10.1007/s11854-007-0017-x.

[6]

A. I. Danilenko and A. del Junco, Cut-and-stack simple weakly mixing map with countably many prime factors, Proc. Amer. Math. Soc., 136 (2008), 2463-2472.  doi: 10.1090/S0002-9939-08-09154-5.

[7]

A. del Junco, A simple map with no prime factors, Israel J. Math., 104 (1998), 301-320.  doi: 10.1007/BF02897068.

[8]

A. del Junco and D. J. Rudolph, A rank-one, rigid, simple, prime map, Ergodic Theory Dynam. Systems, 7 (1987), 229-247.  doi: 10.1017/S0143385700003977.

[9]

A. del JuncoM. Rahe and L. Swanson, Chacon's automorphism has minimal self joinings, J. Analyse Math., 37 (1980), 276-284.  doi: 10.1007/BF02797688.

[10]

S. FerencziC. Holton and L. Q. Zamboni, Joinings of three-interval exchange transformations, Ergodic Theory Dynam. Systems, 25 (2005), 483-502.  doi: 10.1017/S0143385704000811.

[11]

H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.  doi: 10.2307/2373137.

[12] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981. 
[13]

E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.

[14]

E. GlasnerB. Host and D. Rudolph, Simple systems and their higher order self-joinings, Israel J. Math., 78 (1992), 131-142.  doi: 10.1007/BF02801575.

[15]

E. Glasner and B. Weiss, A simple weakly mixing transformation with nonunique prime factors, Amer. J. Math., 116 (1994), 361-375.  doi: 10.2307/2374933.

[16]

P. R. Halmos, In general a measure preserving transformation is mixing, Ann. of Math. (2), 45 (1944), 786-792.  doi: 10.2307/1969304.

[17]

É. JanvresseE. Roy and T. de la Rue, Poisson suspensions and Sushis, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 1301-1334.  doi: 10.24033/asens.2346.

[18]

É. JanvresseA. A. Prikhod'koT. de la Rue and V. V. Ryzhikov, Weak limits of powers of Chacon's automorphism, Ergodic Theory Dynam. Systems, 35 (2015), 128-141.  doi: 10.1017/etds.2013.49.

[19]

É. Janvresse, T. de la Rue and V. Ryzhikov, Around King's rank-one theorems: Flows and $\Bbb Z^n$-actions, Dynamical Systems and Group Actions, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012,143–161. doi: 10.1090/conm/567/11233.

[20]

A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, in Handbook in Dynamical Systems, 1B, Elsevier, 2005,649–743. doi: 10.1016/S1874-575X(06)80036-6.

[21]

J. King, The commutant is the weak closure of the powers, for rank-$1$ transformations, Ergodic Theory Dynam. Systems, 6 (1986), 363-384.  doi: 10.1017/S0143385700003552.

[22]

J. L. King, Joining-rank and the structure of finite rank mixing transformations, J. Analyse Math., 51 (1988), 182-227.  doi: 10.1007/BF02791123.

[23]

J. King, Flat stacks, joining closure and genericity, preprint, available from: http://squash.1gainesville.com/PDF/flatstacks.pdf

[24]

M. LemańczykF. Parreau and E. Roy, Joining primeness and disjointness from infinitely divisible systems, Proc. Amer. Math. Soc., 139 (2011), 185-199.  doi: 10.1090/S0002-9939-2010-10457-4.

[25]

J. LindenstraussG. Olsen and Y. Sternfeld, The Poulsen simplex, Ann. Inst. Fourier (Grenoble), 28 (1978), 91-114.  doi: 10.5802/aif.682.

[26]

W. Parry, Zero entropy of distal and related transformations, in Topological Dynamics (Symp. at Colorado State University, (1967)), Benjamin, New York, 1968,383–389.

[27]

F. Parreau and E. Roy, Prime Poisson suspensions, Ergodic Theory Dynam. Systems, 35 (2015), 2216-2230.  doi: 10.1017/etds.2014.32.

[28]

D. J. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. Analyse Math., 35 (1979), 97-122.  doi: 10.1007/BF02791063.

[29]

V. V. Ryzhikov, Bounded ergodic constructions, disjointness, and weak limits of powers, Trans. Moscow Math. Soc., (2013), 165–171. doi: 10.1090/s0077-1554-2014-00214-4.

[30]

J.-P. Thouvenot, Les systèmes simples sont disjoints de ceux qui sont infiniment divisibles et plongeables dans un flot, Colloq. Math., 84/85 (2000), 481-483.  doi: 10.4064/cm-84/85-2-481-483.

[31]

W. A. Veech, A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341.  doi: 10.1007/BF01667386.

[32]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[33]

R. J. Zimmer, Extensions of ergodic actions and generalized discrete spectrum, Bull. Amer. Math. Soc., 81 (1975), 633-636.  doi: 10.1090/S0002-9904-1975-13770-0.

[34]

R. J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math., 20 (1976), 555-588.  doi: 10.1215/ijm/1256049648.

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