Almost specification ‎and ‎renewality‎ in spacing shifts

Document Type: Research Paper

Authors

1 Department of Pure Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Guilan‎, ‎Iran.

2 Department of Pure Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Guilan‎, ‎Iran.

Abstract

‎Let $(\Sigma_P,\sigma_P)$ be the space of a spacing shifts where $P\subset \mathbb{N}_0=\mathbb{N}\cup\{0\}$ and $\Sigma_P=\{s\in\{0,1\}^{\mathbb{N}_0}: ‎s_i=s_j=1 \mbox{ if } |i-j|\in P \cup\{0\}\}$ and $\sigma_P$ the shift map‎.
‎We will show that $\Sigma_P$ is mixing if and only if it has almost specification property with at least two periodic points‎.
‎Moreover‎, ‎we show that if $h(\sigma_P)=0$‎, ‎then $\Sigma_P$ is almost specified and if $h(\sigma_P)>0$ and $\Sigma_P$ is almost specified‎, ‎then it is weak mixing‎.
‎Also‎, ‎some sufficient conditions for a coded $\Sigma_P$ being renewal or uniquely decipherable is given‎. ‎At last we will show that here are only two conjugacies from a transitive $\Sigma_P$ to a subshift of $\{0,1\}^{\mathbb{N}_0}$‎.

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Main Subjects


D. Ahmadi and M. Dabbaghian, Characterization of spacing shifts with positive topological entropy, Acta Math. Univ. Comenian. 81 (2012), no. 2, 221--226.

J. Banks, T.T.D. Nguyen, P. Oprocha, B. Stanley and B. Trotta, Dynamics of spacing shifts, Discrete Contin. Dyn. Syst. 33 (2013), no. 9, 4207--4232.

J. Banks, P. Oprocha and B. Stanley, Transitive sofic spacing shifts, Discrete Contin. Dyn. Syst. 35 (2015), no. 10, 4734--4764.

V. Bergelson and T. Downarowicz, Large sets of integers and hierarchy of mixing properties of measure-preserving systems, Colloq. Math. 110 (2008), no. 1, 117--150.

V. Bergelson, N. Hindman and R. McCutchen, Notions of size and combinatorial properties of quotient sets in semigroups , Proceedings of the 1998 Topology and Dynamics Conference, Topology Proc. 23 (1998) 23--60.

F. Blanchard and A. Maass, Topics in Symbolic Dynamics and Ar pplications, Cambridge Univ. Press, Cambridge, 2000.

M. Boyle and S. Tuncel, Infinite-to-one codes and Markov measures, Trans. Amer. Math. Soc. 285 (1984), no. 2, 657--684.

H. Furstenberg, Recurrence in Ergodic Theory and Ccombinatorial Number Theory, Princeton Univ. Press, Princeton, NJr , 1981.

J. Goldberger, D. Lind and M. Smorodinsky, The entropies of renewal systems, Israel J. Math. 75 (1991), no. 1, 49--64.

S. Hong and S. Shin, The entropies and periods of renewal systems, Israel J. Math. 172 (2009) 9--27.

M. Kulczycki, D. Kwietniak and P. Oprocha, On almost specification and average shadowing properties, Fund. Math. 224 (2014), no. 3, 241--278.

D. Kwietniak, Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts, Discrete Contin. Dyn. Syst. 33 (2013), no. 6, 2451--2467.

D. Kwietniak and P. Oprocha, A note on the average shadowing property for expansive maps, Topology Appl. 159 (2011), no. 1, 19--27.

K. Lau and A. Zame, On weak mixing of cascades, Math. Systems Theory 6 (1973) 307--311.

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge Univ. Press, Cambridge, 1995.

R. McCutcheon, Three results in recurrence, Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), 349--358, London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995.

K. Petersen, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems 27 (2007) 929--956.

K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc. 190 (1974) 285--299.

P. Walters, On the pseudo orbit tracing property and its relationship to stability, in: The Structure of Attractors in Dynamical Systems (Proc. Conf. North Dakota State Univ. Fargo, ND, 1977), pp. 231--244, Lecture Notes in Math. 668, Springer, Berlin, 1978.