A directional uniformity of periodicpoint distribution and mixing

Richard Miles; Thomas Ward

doi:10.3934/dcds.2011.30.1181

Article Contents

2011, Volume 30, Issue 4: 1181-1189. Doi: 10.3934/dcds.2011.30.1181

This issue Previous Article Some remarks for a modified periodic Camassa-Holm system Next Article New entropy conditions for scalar conservation laws with discontinuous flux

A directional uniformity of periodic point distribution and mixing

Richard Miles^1, and
Thomas Ward^1,

1.
School of Mathematics, University of East Anglia, Norwich, NR4 7TJ

Received: July 2010

Revised: November 2010

Published: October 2011

Abstract Related Papers Cited by

Abstract

For mixing $\mathbb Z^d$-actions generated by commuting automorphisms of a compact abelian group, we investigate the directional uniformity of the rate of periodic point distribution and mixing. When each of these automorphisms has finite entropy, it is shown that directional mixing and directional convergence of the uniform measure supported on periodic points to Haar measure occurs at a uniform rate independent of the direction.

Keywords:

Mathematics Subject Classification: Primary: 37C25, 37C40, 37C35.

Citation:

Related Papers

Cited by

References

[1]	A. Baker, Linear forms in the logarithms of algebraic numbers. IV, Mathematika, 15 (1968), 204-216.doi: 10.1112/S0025579300002588.
[2]	R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 1975.
[3]	M. Boyle and D. Lind, Expansive subdynamics, Trans. Amer. Math. Soc., 349 (1997), 55-102.doi: 10.1090/S0002-9947-97-01634-6.
[4]	M. Einsiedler, M. Kapranov and D. Lind, Non-Archimedean amoebas and tropical varieties, J. Reine Angew. Math., 601 (2006), 139-157.doi: 10.1515/CRELLE.2006.097.
[5]	M. Einsiedler and D. Lind, Algebraic $\mathbb Z^d$-actions of entropy rank one, Trans. Amer. Math. Soc., 356 (2004), 1799-1831.doi: 10.1090/S0002-9947-04-03554-8.
[6]	M. Einsiedler, D. Lind, R. Miles and T. Ward, Expansive subdynamics for algebraic $\mathbb Z^d$-actions, Ergodic Theory Dynam. Systems, 21 (2001), 1695-1729.doi: 10.1017/S014338570100181X.
[7]	Bruce Kitchens and K. Schmidt, Automorphisms of compact groups, Ergodic Theory Dynam. Systems, 9 (1989), 691-735.
[8]	F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), A561-A563.
[9]	D. A. Lind, Dynamical properties of quasihyperbolic toral automorphisms, Ergodic Theory Dynamical Systems, 2 (1982), 49-68.doi: 10.1017/S0143385700009573.
[10]	R. Miles, Zeta functions for elements of entropy rank-one actions, Ergodic Theory Dynam. Systems, 27 (2007), 567-582.doi: 10.1017/S0143385706000794.
[11]	R. Miles and T. Ward, Periodic point data detects subdynamics in entropy rank one, Ergodic Theory Dynam. Systems, 26 (2006), 1913-1930.doi: 10.1017/S014338570600054X.
[12]	R. Miles and T. Ward, Uniform periodic point growth in entropy rank one, Proc. Amer. Math. Soc., 136 (2008), 359-365.doi: 10.1090/S0002-9939-07-09018-1.
[13]	K. Schmidt, "Dynamical Systems of Algebraic Origin," Progress in Mathematics, 128 Birkhäuser Verlag, Basel, 1995.
[14]	T. Ward, The Bernoulli property for expansive $\mathbb Z$² actions on compact groups, Israel J. Math., 79 (1992), 225-249.doi: 10.1007/BF02808217.
[15]	K. R. Yu, Linear forms in p-adic logarithms. II, Compositio Math., 74 (1990), 15-113.