American Institute of Mathematical Sciences

Advanced
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, 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

[1]

Nancy Guelman, Jorge Iglesias, Aldo Portela. Examples of minimal set for IFSs. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5253-5269. doi: 10.3934/dcds.2017227
[2]

Irene Márquez-Corbella, Edgar Martínez-Moro. Algebraic structure of the minimal support codewords set of some linear codes. Advances in Mathematics of Communications, 2011, 5 (2) : 233-244. doi: 10.3934/amc.2011.5.233
[3]

María Isabel Cortez, Samuel Petite. Realization of big centralizers of minimal aperiodic actions on the Cantor set. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2891-2901. doi: 10.3934/dcds.2020153
[4]

Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022
[5]

Alexander Blokh, Michał Misiurewicz. Dense set of negative Schwarzian maps whose critical points have minimal limit sets. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 141-158. doi: 10.3934/dcds.1998.4.141
[6]

Marcin Studniarski. Finding all minimal elements of a finite partially ordered set by genetic algorithm with a prescribed probability. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 389-398. doi: 10.3934/naco.2011.1.389
[7]

Tiago Carvalho, Luiz Fernando Gonçalves. A flow on $ S^2 $ presenting the ball as its minimal set. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4263-4280. doi: 10.3934/dcdsb.2020287
[8]

Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657
[9]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817
[10]

Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231
[11]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519
[12]

Lan Wen. On the preperiodic set. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 237-241. doi: 10.3934/dcds.2000.6.237
[13]

François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262
[14]

Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511
[15]

Gabriel Ponce, Ali Tahzibi, Régis Varão. Minimal yet measurable foliations. Journal of Modern Dynamics, 2014, 8 (1) : 93-107. doi: 10.3934/jmd.2014.8.93
[16]

Virginie Bonnaillie-Noël, Corentin Léna. Spectral minimal partitions of a sector. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 27-53. doi: 10.3934/dcdsb.2014.19.27
[17]

Kristian Bjerklöv, Russell Johnson. Minimal subsets of projective flows. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 493-516. doi: 10.3934/dcdsb.2008.9.493
[18]

Romar dela Cruz, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. On the minimum number of minimal codewords. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020130
[19]

James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667
[20]

Qilin Wang, Liu He, Shengjie Li. Higher-order weak radial epiderivatives and non-convex set-valued optimization problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 465-480. doi: 10.3934/jimo.2018051

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (315)
  • HTML views (172)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy