Beta-Shifts, Their Languages, and Computability
- 88 Downloads
For every real number β>1, the β-shift is a dynamical system describing iterations of the map x ↦ β x mod 1 and is studied intensively in number theory. Each β-shift has an associated language of finite strings of characters; properties of this language are studied for the additional insight they give into the dynamics of the underlying system.
We prove that the language of the β-shift is recursive iff β is a computable real number. That fact yields a precise characterization of the reals: The real numbers β for which we can compute arbitrarily good approximations—hence in particular the numbers for which we can compute their expansion to some base—are precisely those for which there exists a program that decides for any finite sequence of digits whether the sequence occurs as a subword of some element of the β-shift.
While the “only if” part of the proof of the above result is constructive, the “if” part is not. We show that no constructive proof of the “if” part exists. Hence, there exists no algorithm that transforms a program computing arbitrarily good approximations of a real number β into a program deciding the language of the β-shift.
KeywordsComputability theory Dynamical systems Number theory Computable analysis
Unable to display preview. Download preview PDF.
- 3.Berthé, V., Rigo, M.: Abstract numeration systems and tilings. In: Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science (MFCS’05). Lecture Notes in Computer Science, vol. 3618, pp. 131–143. Springer, Berlin (2005) Google Scholar
- 12.Hansel, G., Perrin, D., Simon, I.: Compression and entropy. In: Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science (STACS’92). Lecture Notes in Computer Science, vol. 577, pp. 515–528. Springer, Berlin (1992) Google Scholar
- 14.Hochman, M., Meyerovitch, T.: A characterization of the entropies of multidimensional shifts of finite type. Ann. Math. (2009, to appear) Google Scholar
- 17.Johnson, K.: Beta-shift dynamical systems and their associated languages. Ph.D. Thesis, University of North Carolina (1999) Google Scholar
- 24.Odifreddi, P.: Classical Recursion Theory, vol. I. Studies in Logic and the Foundations of Mathematics, vol. 129. North-Holland, Amsterdam (1989) Google Scholar
- 27.Rogers, H., Jr.: Theory of Recursive Functions and Effective Computability. The MIT Press, Cambridge (1987). Paperback edition Google Scholar
- 30.Sidorov, N.: Arithmetic dynamics. In: Topics in Dynamics and Ergodic Theory. London Mathematical Society Lecture Notes Series, vol. 310, pp. 145–189. London Mathematical Society, London (2003) Google Scholar
- 33.Simonsen, J.: Beta-shifts, their languages and computability. Technical Report, Department of Computer Science, University of Copenhagen (DIKU) (2008) Google Scholar
- 40.Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79. Springer, Berlin (1981) Google Scholar
- 41.Weihrauch, K.: Computable Analysis: An Introduction. Springer, Berlin (1998) Google Scholar