Uniform periodic point growth in entropy rank one

Miles, Richard; Ward, Thomas

doi:10.1090/S0002-9939-07-09018-1

Uniform periodic point growth in entropy rank one
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by Richard Miles and Thomas Ward PDF

Proc. Amer. Math. Soc. 136 (2008), 359-365 Request permission

Abstract:

We show that algebraic dynamical systems with entropy rank one have uniformly exponentially many periodic points in all directions.

References

Alan Baker, Transcendental number theory, Cambridge University Press, London-New York, 1975. MR 0422171, DOI 10.1017/CBO9780511565977
P. E. Blanksby and H. L. Montgomery, Algebraic integers near the unit circle, Acta Arith. 18 (1971), 355–369. MR 296021, DOI 10.4064/aa-18-1-355-369
Mike Boyle and Douglas Lind, Expansive subdynamics, Trans. Amer. Math. Soc. 349 (1997), no. 1, 55–102. MR 1355295, DOI 10.1090/S0002-9947-97-01634-6
V. Chothi, G. Everest, and T. Ward, $S$-integer dynamical systems: periodic points, J. Reine Angew. Math. 489 (1997), 99–132. MR 1461206
Danijela Damjanović and Anatole Katok, Local rigidity of partially hyperbolic actions of $\mathbb Z^k$ and $\mathbb R^k$, $k\ge 2$. I. KAM method and actions on the torus, Preprint.
Manfred Einsiedler, Douglas Lind, Richard Miles, and Thomas Ward, Expansive subdynamics for algebraic ${\Bbb Z}^d$-actions, Ergodic Theory Dynam. Systems 21 (2001), no. 6, 1695–1729. MR 1869066, DOI 10.1017/S014338570100181X
Manfred Einsiedler and Douglas Lind, Algebraic $\Bbb Z^d$-actions of entropy rank one, Trans. Amer. Math. Soc. 356 (2004), no. 5, 1799–1831. MR 2031042, DOI 10.1090/S0002-9947-04-03554-8
Graham Everest and Thomas Ward, Heights of polynomials and entropy in algebraic dynamics, Universitext, Springer-Verlag London, Ltd., London, 1999. MR 1700272, DOI 10.1007/978-1-4471-3898-3
Graham Everest, Richard Miles, Shaun Stevens and Thomas Ward, Orbit-counting in non-hyperbolic dynamical systems, J. Reine Angew. Math., 608 (2007), 155–182.
A. O. Gel′fond, Transcendental and algebraic numbers, Dover Publications, Inc., New York, 1960. Translated from the first Russian edition by Leo F. Boron. MR 0111736
Bruce Kitchens and Klaus Schmidt, Automorphisms of compact groups, Ergodic Theory Dynam. Systems 9 (1989), no. 4, 691–735. MR 1036904, DOI 10.1017/S0143385700005290
François Ledrappier, Un champ markovien peut être d’entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 7, A561–A563 (French, with English summary). MR 512106
D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), no. 3, 461–479. MR 1503118, DOI 10.2307/1968172
D. A. Lind, Dynamical properties of quasihyperbolic toral automorphisms, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 49–68. MR 684244, DOI 10.1017/s0143385700009573
Douglas Lind, Klaus Schmidt, and Tom Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math. 101 (1990), no. 3, 593–629. MR 1062797, DOI 10.1007/BF01231517
Richard Miles, Zeta functions for elements of entropy rank-one actions, Ergodic Theory Dynam. Systems 27 (2007), no. 2, 567–582. MR 2308145, DOI 10.1017/S0143385706000794
Richard Miles, Periodic points of endomorphisms on solenoids and related groups, preprint, 2006.
Richard Miles and Thomas Ward, Periodic point data detects subdynamics in entropy rank one, Ergodic Theory Dynam. Systems 26 (2006), no. 6, 1913–1930. MR 2279271, DOI 10.1017/S014338570600054X
Klaus Schmidt, Dynamical systems of algebraic origin, Progress in Mathematics, vol. 128, Birkhäuser Verlag, Basel, 1995. MR 1345152
Thomas Ward, An uncountable family of group automorphisms, and a typical member, Bull. London Math. Soc. 29 (1997), no. 5, 577–584. MR 1458718, DOI 10.1112/S0024609397003330
T. B. Ward, Almost all $S$-integer dynamical systems have many periodic points, Ergodic Theory Dynam. Systems 18 (1998), no. 2, 471–486. MR 1619569, DOI 10.1017/S0143385798113378
Kun Rui Yu, Linear forms in $p$-adic logarithms. II, Compositio Math. 74 (1990), no. 1, 15–113. MR 1055245
S. A. Juzvinskiĭ, Calculation of the entropy of a group-endomorphism, Sibirsk. Mat. . 8 (1967), 230–239 (Russian). MR 0214726