## Topological Methods in Nonlinear Analysis

### Subshifts, rotations and the specification property

#### Abstract

Let $X=\Sigma_2$ and let $F\colon X\times \mathbb{S}^1\to X\times \mathbb{S}^1$ be a map given by $F(x,t)=(\sigma(x),R_{x_0}(t)),$ where $(\Sigma_2,\sigma)$ denotes the full shift over the alphabet $\{0,1\}$ while $R_0$, $R_1$ are the rotations of the unit circle $\mathbb{S}^1$ by the angles $r_0$ and $r_1$, respectivelly. It was recently proved by X. Wu and G. Chen that if $r_0$ and $r_1$ are irrational, then the system $(X\times \mathbb{S}^1,F)$ has an uncountable distributionally $\delta$-scrambled set $S_\delta$ for every positive $\delta\leq \textrm{diam } X\times \mathbb{S}^1=1$. Moreover, each point in $S_\delta$ is recurrent but not weakly almost periodic (this answeres a question from [Wang et al., Ann. Polon. Math. 82 (2003), 265-272]). We generalize the above result by proving that if $r_0-r_1\in \mathbb R\setminus \mathbb Q$ and $X\subset \Sigma_2$ is a nontrivial subshift with the specification property, then the system $(X\times \mathbb{S}^1,F)$ also has the specification property. As a consequence, there exist a constant $\delta\ge 0$ and a dense Mycielski distributionally $\delta$-scrambled set for $(X\times \mathbb{S}^1,F)$, in which each point is recurrent but not weakly almost periodic.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 46, Number 2 (2015), 799-812.

Dates
First available in Project Euclid: 21 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1458588663

Digital Object Identifier
doi:10.12775/TMNA.2015.077

Mathematical Reviews number (MathSciNet)
MR3494972

Zentralblatt MATH identifier
1362.37027

#### Citation

Mazur, Marcin; Oprocha, Piotr. Subshifts, rotations and the specification property. Topol. Methods Nonlinear Anal. 46 (2015), no. 2, 799--812. doi:10.12775/TMNA.2015.077. https://projecteuclid.org/euclid.tmna/1458588663

#### References

• V.S. Afraimovich and L.P. Shilnikov, Certain global bifurcations connected with the disappearance of a fixed point of saddle-node type, Dokl. Akad. Nauk SSSR 214 (1974), 1281–1284.
• W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math. 79 (1975), 81–92.
• A. Bertrand, Specification, synchronisation, average length, Coding theory and applications (Cachan, 1986), 86–95, Lecture Notes in Comput. Sci., 311, Springer, Berlin, 1988
• R. Bowen, Topological entropy and axiom A, in: “Global Analysis”, Proceedings of Symposia on Pure Mathematics, vol. 14, Amer. Math. Soc., Providence, 1970.
• R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414.
• R.L. Devaney, An introduction to chaotic dynamical systems. Second Edition, in: Addison-Wesley Studies in Nonlinearity, Addison-Wesley, Redwood City, CA, 1989.
• M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Springer–Verlag, Berlin, 1976.
• A. Falcó, The set of periods for a class of crazy maps, J. Math. Anal. Appl. 217 (1998), 546–554.
• S. Glasner and D. Maon, Rigidity in topological dynamics, Ergodic Theory Dynam. Systems 9 (1989), 309–320.
• H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.
• P. Kurka, Topological and Symbolic Dynamics, Cours Spécialisés [Specialized Courses], 11. Société Mathématique de France, Paris, 2003.
• J. Li and P. Oprocha, Shadowing property, weak mixing and regular recurrence, J. Dynam. Differential Equations 25 (2013), 1233–1249.
• J. Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139–147.
• P. Oprocha, Coherent lists and chaotic sets, Discrete Contin. Dyn. Syst. 31 (2011), 797–825.
• ––––, Specification properties and dense distributional chaos, Discrete Contin. Dyn. Syst. 17 (2007), 821–833.
• ––––, Coherent lists and chaotic sets, Discrete Contin. Dyn. Syst. 31 (2011), 797–825.
• P. Oprocha and M. Štefánková, Specification property and distributional chaos almost everywhere, Proc. Amer. Math. Soc. 136 (2008) 3931–3940.
• B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994), 737–754.
• L. Wang, G. Liao, Z. Chen and X. Duan, The set of recurrent points of a continuous self-map on compact metric spaces and strong chaos, Ann. Polon. Math. 82 (2003), 265–272
• H. Wang and L. Wang, The weak specification property and distributional chaos, Nonlinear Anal. 91 (2013), 46–50.
• X. Wu and G. Chen, Non-weakly almost periodic recurrent points and distributionally scrambled sets on $\Sigma\sb 2\times\Bbb{S}\sp 1$, Topology Appl. 162 (2014), 91–99.