Subshifts, rotations and the specification property

Marcin Mazur, Piotr Oprocha



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. \textbf{82} (2003), 265--272]). We generalize the above result by proving that if $r_0-r_1\in \R\setminus \Q$ 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


Distributional chaos; scrambled set; specification property; symbolic dynamics; circle rotation

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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. Falco, 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 Specialises [Specialized Courses], 11. Societe Mathematique 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.

P. Oprocha, Specification properties and dense distributional chaos, Discrete Contin. Dyn. Syst. 17 (2007), 821-833.

P. Oprocha, Coherent lists and chaotic sets, Discrete Contin. Dyn. Syst. 31 (2011), 797-825.

P. Oprocha and M. Stefankova, Specification property and distributional chaos almost everywhere, Proc. Amer. Math. Soc. 136 (2008) 3931-3940.

B. Schweizer and J. Smital, 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 Sum_2times S^1, Topology Appl. 162 (2014), 91-99.


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