-
Previous Article
Cauchy problem for the Kuznetsov equation
- DCDS Home
- This Issue
-
Next Article
Adaptive isogeometric methods with hierarchical splines: An overview
Double minimality, entropy and disjointness with all minimal systems
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 |
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.
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. |
[1] |
Nancy Guelman, Jorge Iglesias, Aldo Portela. Examples of minimal set for IFSs. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5253-5269. doi: 10.3934/dcds.2017227 |
[2] |
Mohsen Tourang, Mostafa Zangiabadi, Chaoqian Li. Generalized minimal Gershgorin set for tensors. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022106 |
[3] |
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 |
[4] |
María Isabel Cortez, Samuel Petite. Realization of big centralizers of minimal aperiodic actions on the Cantor set. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2891-2901. doi: 10.3934/dcds.2020153 |
[5] |
Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations and Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022 |
[6] |
Alexander Blokh, Michał Misiurewicz. Dense set of negative Schwarzian maps whose critical points have minimal limit sets. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 141-158. doi: 10.3934/dcds.1998.4.141 |
[7] |
Marcin Studniarski. Finding all minimal elements of a finite partially ordered set by genetic algorithm with a prescribed probability. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 389-398. doi: 10.3934/naco.2011.1.389 |
[8] |
Tiago Carvalho, Luiz Fernando Gonçalves. A flow on $ S^2 $ presenting the ball as its minimal set. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4263-4280. doi: 10.3934/dcdsb.2020287 |
[9] |
Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657 |
[10] |
Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817 |
[11] |
Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231 |
[12] |
C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial and Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519 |
[13] |
Lan Wen. On the preperiodic set. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 237-241. doi: 10.3934/dcds.2000.6.237 |
[14] |
François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262 |
[15] |
Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511 |
[16] |
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 |
[17] |
Virginie Bonnaillie-Noël, Corentin Léna. Spectral minimal partitions of a sector. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 27-53. doi: 10.3934/dcdsb.2014.19.27 |
[18] |
Kristian Bjerklöv, Russell Johnson. Minimal subsets of projective flows. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 493-516. doi: 10.3934/dcdsb.2008.9.493 |
[19] |
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 |
[20] |
James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]