About entropy

Document Type : Promotional Paper

Author

Faculty of Basic Sciences, Islamic Azad University, Dezful Branch, Dezful, Iran

Abstract

In irreducible subshifts, a word $m$ is synchronizing if whenever $vm$ and $mw$ are admissible words, then $vmw$ is admissible as well. A word $m$ is (left) half (resp. weak) synchronizing, when there is a left transitive ray (resp. a left ray) $x_-$ such that if $x_-m$ and $mw$ are admissible, then $x_-mw$ is also admissible. The resp ective subshifts are called half (resp. weak) synchronized. K. Thomsen considers a synchronized comp onent of a general subshift and investigates the approximation of entropy from inside of this comp onent by some certain SFTs. We, using a rather different approach, show how this result extends to weak synchronized systems.
 
The notion of balanced generator of the Dyke system has b een extended to a general subshift called balanced shift and it has b een shown that they are essentially mixing. All balanced shifts are half synchronized and full shifts are
the only balanced and synchronized subshifts. Also, a formula for the weak half synchronizing entropy of a balanced shift has b een given.

Keywords

Main Subjects

References

[1] D. Ahmadi Dastjerdi and M. Shahamat, Balanced Shifts, Communications in Mathematics and Applications, 11 (2020) 415–424.
[2] D. Ahmadi Dastjerdi and M. Shahamat, Beyond Half Synchronized Systems, Majlesi Journal of Telecommunication Devices, 8 (2019) 69–73.
[3] D. Ahmadi Dastjerdi and M. Shahamat, Kreiger Graphs and Fischer Covers vs Dynamical Properties , J. Adv. Math. Comput. Sci, 32 (2019) 1–12.
[4] F. Blanchard and G. Hansel, Systèmes codés, Theoret. Comput. Sci., 44 (1986) 17–49.
[5] V. Climenhaga and D. J. Thompson, Intrinsic ergodicity beyond specification: β -shifts, S -gap shifts, and their factors, Israel J. Math., 192 (2012) 785–817.
[6] D. Fiebig and U. Fiebig, Covers for coded systems, Amer. Math. Soc., 135 (1992) 139–179.
[7] K. Ch. Johnson, Beta-shift dynamical systems and their associated languages, The University of North Carolina at Chapel Hill, ProQuest LLC, Ann Arbor, MI, 1999 124 pp.
[8] U. Jung, on the existence of open and bi-continuous codes, Trans. Amer. Math. Soc., 363 (2011) 1399–1417.
[9] B. Kitchens, Continuity properties of factor maps in ergodic theory, The University of North Carolina at Chapel Hill, ProQuest LLC, Ann Arbor, MI, 1981 29 pp.
[10] D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995.
[11] T. Meyerovitvh, Tail invariant measures of the Dyke-shift and non-sofic systems, M.Sc. Thesis, Tel-Aviv university, 2004.
[12] Z. H. Nitecki, Topological entropy and the preimage structure of maps, Real Anal. Exchange, 29 (2003/04) 9–41.
[13] M. Shahamat, Synchronized components of a subshift, J. Korean Math. Soc., 59 (2022) 1–12.
[14] M. Shahamat, Strong Synchronized System, Journal of Mathematical Extension Vol. 16, No. 6, (2022) (4)1-16.
[15] M. Shahamat, Infinite minimal half synchronizing, Journal of Mahani Mathematical Research Center, preprint.
[16] M. Shahamat, D. Ahmadi Dastjerdi and B. Panbehkar, Minimally Generated Subshifts, J. Math. and Appl., 1 (2021) 18–24.
[17] K. Thomsen, on the ergodic theory of synchronized systems, Ergod. Th. Dynam. Sys., 356 (2006) 1235–1256.
[18] K. Thomsen, On the structure of a sofic shift space, Amer. Math. Soc., 356 (2004) 3557–3619.
[19] T. Meyerovitch, Tail invariant measures of the Dyck-shift and non-sofic systems, M. Sc. Thesis, Tel-Aviv University, 2004.
Mathematics and Society
Volume 6, Issue 4 - Serial Number 4
December 2021
Pages 7-27

Files

History

  • Receive Date: 08 February 2022
  • Revise Date: 05 August 2022
  • Accept Date: 17 August 2022
  • Publish Date: 20 February 2022

Share

How to cite

Statistics