Abstract
We consider biinfinite sequences on a finite alphabet of digits which satisfy a constraint of finite type. Such sequences are perturbed by adding a 1 in position 0. The odometer is the function which transforms the initial sequence into an admissible sequence equivalent to the perturbed one. It is shown that the odometer can be realized by an on-line finite automaton when the constraint is linked to numeration in base β, where β is a Pisot number satisfying the equation βm = t1βm-1+⋯+t m, where t 1 ≥ t 2 ≥ . . . t m ≥1 are integers.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Avizienis, Signed-digit number representations for fast parallel arithmetic. IRE Transactions on electronic computers 10 (1961), 389–400.
A. Bertrand-Mathis, Comment écrire les nombres entiers dans une base qui n’est pas entière. Acta Math. Acad. Sci. Hungar. 54 (1989), 237–241.
A. Brauer, On algebraic equations with all but one root in the interior of the unit circle. Math. Nachr. 4 (1951), 250–257.
C.Y. Chow and J.E. Robertson, Logical design of a redundant binary adder. Proc. 4th Symposium on Computer Arithmetic, I.E.E.E. Computer Society Press (1978), 109–115.
S. Eilenberg, Automata, Languages and Machines, vol. A, Academic Press, 1974.
Ch. Frougny, Representation of numbers and finite automata. Math. Systems Theory 25 (1992), 37–60.
Ch. Frougny, On the sequentiality of the successor function. Inform. and Computation 139 (1997), 17–38.
Ch. Frougny, On-line digit set conversion in real base. Theor. Comp. Sci. 292 (2002), 221–235.
Ch. Frougny, On-line odometers, manuscript.
Ch. Frougny and J. Sakarovitch, Synchronisation déterministe des automates à délai borné. Theor. Comp. Sci. 191 (1998), 61–77.
Ch. Frougny and B. Solomyak, Finite beta-expansions. Ergodic Theory & Dynamical Systems 12 (1992), 713–723.
P. Grabner, P. Liardet, R. Tichy, Odometers and systems of numeration. Acta Arithmetica LXXX.2 (1995), 103–123.
S. Ito and Y. Sano, Substitutions, atomic surfaces, and periodic beta expansions. Analytic Number Theory, C. Jia and K. Matsumoto eds., Developments in Mathematics, Volume 6, 2002.
D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, 1995.
M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, 2002.
J.-M. Muller, Some characterizations of functions computable in on-line arithmetic. I.E.E.E. Trans. on Computers, 43 (1994), 752–755.
W. Parry, On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.
A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493.
K. Schmidt, Algebraic coding of expansive group automorphisms and two-sided beta-shifts. Monatsh. Math. 129 (2000), 37–61.
N. Sidorov and A. Vershik, Ergodic properties of the Erdős measure, the entropy of the golden shift, and related problems. Monatsh. Math. 126 (1998), 215–261.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Frougny, C. (2003). On-Line Odometers for Two-Sided Symbolic Dynamical Systems. In: Ito, M., Toyama, M. (eds) Developments in Language Theory. DLT 2002. Lecture Notes in Computer Science, vol 2450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45005-X_36
Download citation
DOI: https://doi.org/10.1007/3-540-45005-X_36
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40431-6
Online ISBN: 978-3-540-45005-4
eBook Packages: Springer Book Archive