A directional uniformity of periodic point distribution and mixing

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November 2011, 30(4): 1181-1189. doi: 10.3934/dcds.2011.30.1181

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, United Kingdom

Received July 2010 Revised November 2010 Published May 2011

Abstract
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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: Rate of mixing, commuting transformations., equidistribution of periodic points, directional uniformity.

Mathematics Subject Classification: Primary: 37C25, 37C40, 37C3.

Citation: Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181

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.

show all references

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.

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