Document Type: Research Paper


Department of Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎Tarbiat Modares University‎, ‎Tehran 14115-134‎, ‎Iran


We extend the results of Walters on the uniqueness of invariant measures with maximal entropy on compact groups to an arbitrary locally compact group. We show that the maximal entropy is attained at the left Haar measure and the measure of maximal entropy is unique.


Main Subjects

[1] J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs 50, American Mathematical Society, Providence, 1997.

[2] M. Amini, Invariant measures on non compact dynamics, preprint.

[3] K. R. Berg, Convolution of invariant measures, maximal entropy, Math. Syst. Theory. 3 (1969), 146-151.

[4] L. W. Goodwyn, Comparing topological entropy with measure-theoretic entropy, Amer. J. Math. 94 (1972), 366-388.

[5] B. M. Gurevic, Topological entropy of denumerable Markov chains, Soviet Math. Dokl. 10 (4) (1969), 911-915.

[6] N. Krylov, N. Bogolioubov, La th´eorie g´en´erale de la mesure dans son application ´al’´etude des syst´emes
dynamiques de la m´ ecanique non lin´ eaire, Ann. Math. 38 (1937), 65-113.

[7] V. V. Nemycki˘i, V. V. Stepanov, Qualitative Theory of Differential Equations, 2nd Ed., Moscow, 1949.

[8] J. C. Oxtoby, S. M. Ulam, On the existence of a measure invariant under a transformation, Annals Math. 40 (1939), 560-566.

[9] K. Sakai, Various shadowing properties for positively expansive maps, Topol. Appl. 131 (2003), 15-31.

[10] M. Viana, K. Oliveira, Foundations of Ergodic Theory, Cambridge Studies in Advanced Mathematics 151, Cambridge University Press, 2016.

[11] P. Walters, Ergodic theory-Introductory lectures, Lecture Notes in Math. 458, Springer-Verlag, New York, 1975.

[12] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121-153.

[13] P. Walters, Some transformations having a unique measure with maximal entropy, Proc. London Math. Soc. (3) 28 (1974), 500-516.

[14] B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc. 76 (1970), 1266-1269.