August  2019, 39(8): 4331-4344. doi: 10.3934/dcds.2019175

Follower, predecessor, and extender set sequences of $ \beta $-shifts

Department of Mathematics, University of Denver, C.M.Knudson Hall, Room 300, 2390 S. York St, Denver, CO 80208, USA

* Corresponding author: Thomas French.

Received  November 2017 Revised  January 2019 Published  May 2019

Given a one-dimensional shift $ X $ and a word $ v $ in the language of $ X $, the follower set of $ v $ is the set of all finite words which can legally follow $ v $ in some point of $ X $. The predecessor set of $ v $ is the set of all finite words which can legally precede $ v $ in some point of $ X $. We construct the follower set sequence of $ X $ by recording, for each $ n $, the number of distinct follower sets of words of length $ n $ in $ X $. We construct the predecessor set sequence of $ X $ by recording, for each $ n $, the number of distinct predecessor sets of words of length $ n $ in $ X $. Extender sets are a generalization of follower sets (see [6]), and we define the extender set sequence similarly. In this paper, we examine achievable differences in limiting behavior of follower, predecessor, and extender set sequences. This is done through the classical $ \beta $-shifts, first introduced in [10]. We show that the follower set sequences of $ \beta $-shifts must grow at most linearly in $ n $, while the predecessor and extender set sequences may demonstrate exponential growth rate in $ n $, depending on choice of $ \beta $.

Citation: Thomas French. Follower, predecessor, and extender set sequences of $ \beta $-shifts. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4331-4344. doi: 10.3934/dcds.2019175
References:
[1]

F. Blanchard, $\beta$-expansions and symbolic dynamics, Theoret. Comput. Sci., 65 (1989), 131-141.  doi: 10.1016/0304-3975(89)90038-8.  Google Scholar

[2]

D. P. Chi and D. Kwon, Sturmian words, $\beta$-shifts, and transcendence, Theoret. Comput. Sci., 321 (2004), 395-404.  doi: 10.1016/j.tcs.2004.03.035.  Google Scholar

[3]

T. French, Characterizing follower and extender set sequences, Dyn. Syst., 31 (2016), 293-310.  doi: 10.1080/14689367.2015.1111865.  Google Scholar

[4]

T. French, N. Ormes and R. Pavlov, Subshifts with slowly growing numbers of follower sets, in Ergodic theory, dynamical systems, and the continuing in uence of John C. 506 Oxtoby, volume 678 of Contemp. Math., Amer. Math. Soc., (2016), 175–186.  Google Scholar

[5]

T. French and R. Pavlov, Follower, predecessor, and extender entropies, Monatsh. Math., 188 (2019), 495–510, arXiv: 1711.07515. doi: 10.1007/s00605-018-1224-5.  Google Scholar

[6]

S. Kass and K. Madden, A sufficient condition for non-soficness of higher-dimensional subshifts, Proc. Amer. Math. Soc., 141 (2013), 3803-3816.  doi: 10.1090/S0002-9939-2013-11646-1.  Google Scholar

[7] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar
[8]

N. Ormes and R. Pavlov, Extender sets and multidimensional subshifts, Ergodic Theory Dynam. Systems, 36 (2016), 908-923.  doi: 10.1017/etds.2014.71.  Google Scholar

[9]

W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar, 11 (1960), 401-416.  doi: 10.1007/BF02020954.  Google Scholar

[10]

A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, 8 (1957), 477-493.  doi: 10.1007/BF02020331.  Google Scholar

show all references

References:
[1]

F. Blanchard, $\beta$-expansions and symbolic dynamics, Theoret. Comput. Sci., 65 (1989), 131-141.  doi: 10.1016/0304-3975(89)90038-8.  Google Scholar

[2]

D. P. Chi and D. Kwon, Sturmian words, $\beta$-shifts, and transcendence, Theoret. Comput. Sci., 321 (2004), 395-404.  doi: 10.1016/j.tcs.2004.03.035.  Google Scholar

[3]

T. French, Characterizing follower and extender set sequences, Dyn. Syst., 31 (2016), 293-310.  doi: 10.1080/14689367.2015.1111865.  Google Scholar

[4]

T. French, N. Ormes and R. Pavlov, Subshifts with slowly growing numbers of follower sets, in Ergodic theory, dynamical systems, and the continuing in uence of John C. 506 Oxtoby, volume 678 of Contemp. Math., Amer. Math. Soc., (2016), 175–186.  Google Scholar

[5]

T. French and R. Pavlov, Follower, predecessor, and extender entropies, Monatsh. Math., 188 (2019), 495–510, arXiv: 1711.07515. doi: 10.1007/s00605-018-1224-5.  Google Scholar

[6]

S. Kass and K. Madden, A sufficient condition for non-soficness of higher-dimensional subshifts, Proc. Amer. Math. Soc., 141 (2013), 3803-3816.  doi: 10.1090/S0002-9939-2013-11646-1.  Google Scholar

[7] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar
[8]

N. Ormes and R. Pavlov, Extender sets and multidimensional subshifts, Ergodic Theory Dynam. Systems, 36 (2016), 908-923.  doi: 10.1017/etds.2014.71.  Google Scholar

[9]

W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar, 11 (1960), 401-416.  doi: 10.1007/BF02020954.  Google Scholar

[10]

A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, 8 (1957), 477-493.  doi: 10.1007/BF02020331.  Google Scholar

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