An Alternative Proof for the Topological Entropy of the Motzkin Shift
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Description
A Motzkin shift is a mathematical model for constraints
on genetic sequences. In terms of the theory of symbolic dynamics,
the Motzkin shift is nonsofic, and therefore, we cannot use the Perron-
Frobenius theory to calculate its topological entropy. The Motzkin
shift M(M,N) which comes from language theory, is defined to be the
shift system over an alphabet A that consists of N negative symbols,
N positive symbols and M neutral symbols. For an x in the full shift,
x will be in the Motzkin subshift M(M,N) if and only if every finite
block appearing in x has a non-zero reduced form. Therefore, the
constraint for x cannot be bounded in length. K. Inoue has shown that
the entropy of the Motzkin shift M(M,N) is log(M + N + 1). In this
paper, a new direct method of calculating the topological entropy of
the Motzkin shift is given without any measure theoretical discussion.
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References
- D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge Univ. Press, 1995.
- K. Inoue, The Zeta function, Periodic Points and Entropies of the Motzkin Shift, arXiv:math/0602100v3, (math.DS), 2006.
- P. B. Kitchens, Symbolic Dynamics. One-sided, two-sided and countable state Markov shifts, Berlin: Universitext, Springer-Verlag, 1998.
- Z. H. Nitecki, Topological entropy and the preimage structure of maps, Real Analysis Exchange, pp 9-42, 2003/2004.
- W. Krieger, On the uniqeness of equilibrium state, Mathematical system theory, Vol. 8, pp 97-104. 1974.