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January  2019, 39(1): 263-275. doi: 10.3934/dcds.2019011

Double minimality, entropy and disjointness with all minimal systems

Piotr Oprocha 1,2,

1. 

AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland
2. 

National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic

Received  December 2017 Revised  July 2018 Published  October 2018

In this paper we propose a new sufficient condition for disjointness with all minimal systems.

Using proposed approach we construct a transitive dynamical system $(X,T)$ disjoint with every minimal system and such that the set of transfer times $N(x,U)$ is not an $\text{IP}^*$-set for some nonempty open set $U\subset X$ and every $x∈ X$. This example shows that the new condition sharpens sufficient conditions for disjointness below previous bounds. In particular our approach does not rely on distality of points or sets.

Keywords: Weak mixing, disjoint, doubly minimal, IP-set, minimal set.
Mathematics Subject Classification: Primary: 37B20; Secondary: 37B05, 37B40.
Citation: Piotr Oprocha. Double minimality, entropy and disjointness with all minimal systems. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 263-275. doi: 10.3934/dcds.2019011
References:
[1]

E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433.  doi: 10.1088/0951-7715/16/4/313.

[2]

F. Blanchard, A disjointness theorem involving topological entropy, Bull. Soc. Math. France, 121 (1993), 465-478.  doi: 10.24033/bsmf.2216.

[3]

F. Blanchard, B. Host and S. Ruette, Asymptotic pairs in positive-entropy systems, Ergodic Theory Dynam. Systems, 22 (2002), 671-686.  doi: 10.1017/S0143385702000342.

[4]

P. Dong, S. Shao and X. Ye, Product recurrent properties, disjointness and weak disjointness, Israel J. Math., 188 (2012), 463-507.  doi: 10.1007/s11856-011-0128-z.

[5]

F. Falniowski, M. Kulczycki, D. Kwietniak and J. Li, Two results on entropy, chaos and independence in symbolic dynamics, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3487-3505.  doi: 10.3934/dcdsb.2015.20.3487.

[6]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.

[7]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures. Princeton University Press, Princeton, N.J., 1981.

[8]

H. Furstenberg, H. Keynes and L. Shapiro, Prime flows in topological dynamics, Israel J. Math., 14 (1973), 26-38.  doi: 10.1007/BF02761532.

[9]

E. Glanser and B. Weiss, On the interplay between measurable and topological dynamics, Handbook of Dynamical Systems, vol. 1B, Elsevier, (2006), 597-648. doi: 10.1016/S1874-575X(06)80035-4.

[10]

E. Glanser and B. Weiss, On doubly minimal systems and a question regarding product recurrence, preprint, arXiv: 1508.02817, 2015.

[11]

C. Grillenberger, Constructions of strictly ergodic systems. Ⅱ. K-Systems, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 25 (1972/73), 335-342.  doi: 10.1007/BF00537162.

[12]

K. Haddad and W. Ott, Recurrence in pairs, Ergodic Theory Dynam. Systems, 28 (2008), 1135-1143.  doi: 10.1017/S0143385707000600.

[13]

W. Huang and X. Ye, Dynamical systems disjoint from any minimal system, Trans. Amer. Math. Soc., 357 (2005), 669-694.  doi: 10.1090/S0002-9947-04-03540-8.

[14]

W. Huang and X. Ye, A local variational relation and applications, Israel J. Math., 151 (2006), 237-279.  doi: 10.1007/BF02777364.

[15]

W. Huang and X. Ye, A note on double minimality, Commun. Math. Stat., 3 (2015), 57-61.  doi: 10.1007/s40304-015-0051-4.

[16]

J. L. King, A map with topological minimal self-joinings in the sense of del Junco, Ergodic Theory Dynam. Systems, 10 (1990), 745-761.  doi: 10.1017/S0143385700005873.

[17]

P. Kurka, Topological and Symbolic Dynamics, Cours Spécialisés [Specialized Courses], 11. Société Mathématique de France, Paris, 2003.

[18]

J. Li, K. Yan and X. Ye, Recurrence properties and disjointness on the induced spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1059-1073.  doi: 10.3934/dcds.2015.35.1059.

[19]

J. Li, P. Oprocha, X. Ye and R. Zhang, When all closed subsets are recurrent?, Erg. Th. Dynam. Syst., 37 (2017), 2223-2254.  doi: 10.1017/etds.2016.5.

[20]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.

[21]

E. Lindenstrauss, Lowering topological entropy, J. Anal. Math., 67 (1995), 231-267.  doi: 10.1007/BF02787792.

[22]

P. Oprocha, Weak mixing and product recurrence, Ann. Inst. Fourier (Grenoble), 60 (2010), 1233-1257.  doi: 10.5802/aif.2553.

[23]

P. Oprocha, Minimal systems and distributionally scrambled sets, Bull. Soc. Math. France, 140 (2012), 401-439.  doi: 10.24033/bsmf.2631.

[24]

P. Oprocha and G. Zhang, On weak product recurrence and synchronization of return times, Adv. Math., 244 (2013), 395-412.  doi: 10.1016/j.aim.2013.05.006.

[25]

K. E. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc., 24 (1970), 278-280.  doi: 10.1090/S0002-9939-1970-0250283-7.

[26]

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

[27]

B. Weiss, Multiple recurrence and doubly minimal systems, Topological Dynamics and Applications (Minneapolis, MN, 1995), 189-196, Contemp. Math., 215, Amer. Math. Soc., Providence, RI, 1998. doi: 10.1090/conm/215/02940.

show all references

References:
[1]

E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433.  doi: 10.1088/0951-7715/16/4/313.

[2]

F. Blanchard, A disjointness theorem involving topological entropy, Bull. Soc. Math. France, 121 (1993), 465-478.  doi: 10.24033/bsmf.2216.

[3]

F. Blanchard, B. Host and S. Ruette, Asymptotic pairs in positive-entropy systems, Ergodic Theory Dynam. Systems, 22 (2002), 671-686.  doi: 10.1017/S0143385702000342.

[4]

P. Dong, S. Shao and X. Ye, Product recurrent properties, disjointness and weak disjointness, Israel J. Math., 188 (2012), 463-507.  doi: 10.1007/s11856-011-0128-z.

[5]

F. Falniowski, M. Kulczycki, D. Kwietniak and J. Li, Two results on entropy, chaos and independence in symbolic dynamics, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3487-3505.  doi: 10.3934/dcdsb.2015.20.3487.

[6]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.

[7]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures. Princeton University Press, Princeton, N.J., 1981.

[8]

H. Furstenberg, H. Keynes and L. Shapiro, Prime flows in topological dynamics, Israel J. Math., 14 (1973), 26-38.  doi: 10.1007/BF02761532.

[9]

E. Glanser and B. Weiss, On the interplay between measurable and topological dynamics, Handbook of Dynamical Systems, vol. 1B, Elsevier, (2006), 597-648. doi: 10.1016/S1874-575X(06)80035-4.

[10]

E. Glanser and B. Weiss, On doubly minimal systems and a question regarding product recurrence, preprint, arXiv: 1508.02817, 2015.

[11]

C. Grillenberger, Constructions of strictly ergodic systems. Ⅱ. K-Systems, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 25 (1972/73), 335-342.  doi: 10.1007/BF00537162.

[12]

K. Haddad and W. Ott, Recurrence in pairs, Ergodic Theory Dynam. Systems, 28 (2008), 1135-1143.  doi: 10.1017/S0143385707000600.

[13]

W. Huang and X. Ye, Dynamical systems disjoint from any minimal system, Trans. Amer. Math. Soc., 357 (2005), 669-694.  doi: 10.1090/S0002-9947-04-03540-8.

[14]

W. Huang and X. Ye, A local variational relation and applications, Israel J. Math., 151 (2006), 237-279.  doi: 10.1007/BF02777364.

[15]

W. Huang and X. Ye, A note on double minimality, Commun. Math. Stat., 3 (2015), 57-61.  doi: 10.1007/s40304-015-0051-4.

[16]

J. L. King, A map with topological minimal self-joinings in the sense of del Junco, Ergodic Theory Dynam. Systems, 10 (1990), 745-761.  doi: 10.1017/S0143385700005873.

[17]

P. Kurka, Topological and Symbolic Dynamics, Cours Spécialisés [Specialized Courses], 11. Société Mathématique de France, Paris, 2003.

[18]

J. Li, K. Yan and X. Ye, Recurrence properties and disjointness on the induced spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1059-1073.  doi: 10.3934/dcds.2015.35.1059.

[19]

J. Li, P. Oprocha, X. Ye and R. Zhang, When all closed subsets are recurrent?, Erg. Th. Dynam. Syst., 37 (2017), 2223-2254.  doi: 10.1017/etds.2016.5.

[20]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.

[21]

E. Lindenstrauss, Lowering topological entropy, J. Anal. Math., 67 (1995), 231-267.  doi: 10.1007/BF02787792.

[22]

P. Oprocha, Weak mixing and product recurrence, Ann. Inst. Fourier (Grenoble), 60 (2010), 1233-1257.  doi: 10.5802/aif.2553.

[23]

P. Oprocha, Minimal systems and distributionally scrambled sets, Bull. Soc. Math. France, 140 (2012), 401-439.  doi: 10.24033/bsmf.2631.

[24]

P. Oprocha and G. Zhang, On weak product recurrence and synchronization of return times, Adv. Math., 244 (2013), 395-412.  doi: 10.1016/j.aim.2013.05.006.

[25]

K. E. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc., 24 (1970), 278-280.  doi: 10.1090/S0002-9939-1970-0250283-7.

[26]

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

[27]

B. Weiss, Multiple recurrence and doubly minimal systems, Topological Dynamics and Applications (Minneapolis, MN, 1995), 189-196, Contemp. Math., 215, Amer. Math. Soc., Providence, RI, 1998. doi: 10.1090/conm/215/02940.

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