On the absence of invariant measures

with locally maximal entropy

for a class of shifts of finite type

Authors:
E. Arthur Robinson Jr. and Ayse A. Sahin

Journal:
Proc. Amer. Math. Soc. **127** (1999), 3309-3318

MSC (1991):
Primary 28D15; Secondary 28D20

DOI:
https://doi.org/10.1090/S0002-9939-99-05215-6

Published electronically:
May 13, 1999

MathSciNet review:
1646203

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that for a class of shifts of finite type, , any invariant measure which is not a measure of maximal entropy can be perturbed a small amount in the weak* topology to an invariant measure of higher entropy. Namely, there are no invariant measures which are strictly local maxima for the entropy function.

**1.**R. M. Burton and J. E. Steif. Non-uniqueness of measures of maximal entropy for subshifts of finite type.*Ergodic Theory Dynam. Systems*, 14(2):213-235, 1994. MR**95f:28023****2.**R. M. Burton and J. E. Steif. Some -d symbolic dynamical systems: entropy and mixing. In*Ergodic theory of actions (Warwick, 1993-1994)*, volume 228 of*London Math. Soc. Lecture Note Ser.*, pages 297-305. Cambridge Univ. Press, Cambridge, 1996. MR**97e:58066****3.**N. G. Markley and M. E. Paul. Maximal measures and entropy for subshifts of finite type. In*Classical mechanics and dynamical systems (Medford, Mass., 1979)*, volume 70 of*Lecture Notes in Pure and Appl. Math., 70*, pages 135-157. Dekker, New York, 1981. MR**83c:54059****4.**M. Misiurewicz. A short proof of the variational principle for a action on a compact space. In*International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975)*, pages 147-157. Astérisque, No. 40. Soc. Math. France, Paris, 1976. MR**56:3250****5.**D. S. Ornstein and B. Weiss. Entropy and isomorphism theorems for actions of amenable groups.*J. Analyse Math.*, 48:1-141, 1987. MR**88j:28014****6.**D. J. Rudolph.*Fundamentals of measurable dynamics*. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, 1990. MR**92e:28006****7.**K. Schmidt.*Dynamical systems of algebraic origin*, volume 128 of*Progress in Mathematics*. Birkhäuser Verlag, Basel, 1995. MR**97c:28041**

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Additional Information

**E. Arthur Robinson Jr.**

Affiliation:
Department of Mathematics, George Washington University, Washington, DC 20052

Email:
robinson@math.gwu.edu

**Ayse A. Sahin**

Affiliation:
Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105

Email:
sahin@plains.nodak.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-05215-6

Keywords:
Ergodic theory,
$\mathbb Z^d$ actions,
entropy,
symbolic dynamics

Received by editor(s):
February 6, 1998

Published electronically:
May 13, 1999

Additional Notes:
The research of the first author was partially supported by the NSF under grant number DMS 9303498.

The research of the second author was partially supported by the NSF under grant number DMS 9501103.

Communicated by:
Michael Handel

Article copyright:
© Copyright 1999
American Mathematical Society