# American Institute of Mathematical Sciences

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

## A directional uniformity of periodic point distribution and mixing

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

Received  July 2010 Revised  November 2010 Published  May 2011

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

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