Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Boundaries of Markov partitions


Authors: Jonathan Ashley, Bruce Kitchens and Matthew Stafford
Journal: Trans. Amer. Math. Soc. 333 (1992), 177-201
MSC: Primary 58F15; Secondary 28D05, 54H20, 58F11
DOI: https://doi.org/10.1090/S0002-9947-1992-1073772-3
MathSciNet review: 1073772
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The core of a Markov partition is the nonwandering set of the map restricted to the boundary of the partition. We show that the core of a Markov partition is always a finitely presented system. Then we show that every one sided sofic system occurs as the core of a Markov partition for an $ n$-fold covering map on the circle and every two sided sofic system occurs as the core of a Markov partition for a hyperbolic automorphism of the two dimensional torus.


References [Enhancements On Off] (What's this?)

  • [AGW] R. Adler, W. Goodwyn, and B. Weiss, Equivalence of topological Markov shifts, Israel J. Math. 27 (1977), 49-63. MR 0437715 (55:10639)
  • [AW] R. Adler and B. Weiss, Similarity of automorphisms of the torus, Mem. Amer. Math. Soc., no. 98, 1965. MR 0257315 (41:1966)
  • [AM] R. Adler and B. Marcus, Topological entropy and the equivalence of dynamical systems, Mem. Amer. Math. Soc., no. 219, 1979. MR 533691 (83h:28027)
  • [Bo1] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, 1975. MR 0442989 (56:1364)
  • [Bo2] -, On Axiom A diffeomorphisms, CBMS Regional Conf. Ser. in Math., no. 35, Amer. Math. Soc., Providence, R.I., 1978. MR 0482842 (58:2888)
  • [Bo3] -, Markov partitions are not smooth, Proc. Amer. Math. Soc. 71 (1978), 130-132. MR 0474415 (57:14055)
  • [CP] E. Coven and M. Paul, Endomorphisms of irreducible subshifts of finite type, Math. Systems Theory 8 (1974), 167-175. MR 0383378 (52:4259)
  • [CR] E. Coven and W. Reddy, Positively expansive maps of compact manifolds.
  • [F] D. Fried, Finitely presented dynamical systems, Ergodic Theory Dynamical Systems 7 (1987), 489-507. MR 922362 (89h:58157)
  • [H] G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory 3 (1969), 320-375. MR 0259881 (41:4510)
  • [Kr1] W. Krieger, On sofic systems. I, Israel J. Math. 48 (1984), 305-330. MR 776312 (86j:54074)
  • [Kr2] -, On the subsystems of topological Markov chains, Ergodic Theory Dynamical Systems 2 (1982), 195-202. MR 693975 (85b:28020)
  • [M] B. Marcus, Factors and extensions of full shifts, Monatsh. Math. 88 (1979), 239-247. MR 553733 (81g:28023)
  • [S] M. Shub, Global stability of dynamical systems, Springer-Verlag, 1987. MR 869255 (87m:58086)
  • [St] M. Stafford, Markov partitions for expanding maps of the circle, Trans. Amer. Math. Soc. 324 (1991), 385-403. MR 1049617 (91f:58053)
  • [W] B. Weiss, Subshifts of finite type and sofic systems, Monatsh. Math. 77 (1973), 462-478. MR 0340556 (49:5308)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F15, 28D05, 54H20, 58F11

Retrieve articles in all journals with MSC: 58F15, 28D05, 54H20, 58F11


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1073772-3
Keywords: Markov partitions, sofic systems
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society