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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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

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