Group automorphisms with few and with many periodic points
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Abstract:
For each $C\in [0,\infty ]$ a compact group automorphism $T:X\to X$ is constructed with the property that \[ \frac {1}{n}\log \vert \{x\in X\mid T^n(x)=x\}\vert \longrightarrow C. \] This may be interpreted as a combinatorial analogue of the (still open) problem of whether compact group automorphisms with any given topological entropy exist.References
- R. Bowen and O. E. Lanford III, Zeta functions of restrictions of the shift transformation, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 43–49. MR 0271401
- V. Chothi, G. Everest, and T. Ward, $S$-integer dynamical systems: periodic points, J. Reine Angew. Math. 489 (1997), 99–132. MR 1461206
- James W. England and Roy L. Smith, The zeta function of automorphisms of solenoid groups, J. Math. Anal. Appl. 39 (1972), 112–121. MR 307280, DOI 10.1016/0022-247X(72)90228-4
- G. Everest, A. J. van der Poorten, Y. Puri, and T. Ward, Integer sequences and periodic points, J. Integer Seq. 5 (2002), no. 2, Article 02.2.3, 10. MR 1938222
- Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence sequences, Mathematical Surveys and Monographs, vol. 104, American Mathematical Society, Providence, RI, 2003. MR 1990179, DOI 10.1090/surv/104
- 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
- D. R. Heath-Brown, Zero-free regions for Dirichlet $L$-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3) 64 (1992), no. 2, 265–338. MR 1143227, DOI 10.1112/plms/s3-64.2.265
- 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, The structure of skew products with ergodic group automorphisms, Israel J. Math. 28 (1977), no. 3, 205–248. MR 460593, DOI 10.1007/BF02759810
- 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
- D. A. Lind and T. Ward, Automorphisms of solenoids and $p$-adic entropy, Ergodic Theory Dynam. Systems 8 (1988), no. 3, 411–419. MR 961739, DOI 10.1017/S0143385700004545
- U. V. Linnik, On the least prime in an arithmetic progression. I. The basic theorem, Rec. Math. [Mat. Sbornik] N.S. 15(57) (1944), 139–178 (English, with Russian summary). MR 0012111
- P. Moss, Algebraic realizability problems, Ph.D. thesis, The University of East Anglia, 2003.
- G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. II, Revised and enlarged translation by C. E. Billigheimer of the fourth German edition, Springer Study Edition, Springer-Verlag, New York-Heidelberg, 1976. Theory of functions, zeros, polynomials, determinants, number theory, geometry. MR 0465631, DOI 10.1007/978-1-4757-6292-1
- Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seq. 4 (2001), no. 2, Article 01.2.1, 18. MR 1873399
- 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
Additional Information
- Thomas Ward
- Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom
- MR Author ID: 180610
- Email: t.ward@uea.ac.uk
- Received by editor(s): August 16, 2003
- Published electronically: August 10, 2004
- Communicated by: Michael Handel
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 91-96
- MSC (2000): Primary 37C35, 22D40, 11N13
- DOI: https://doi.org/10.1090/S0002-9939-04-07626-9
- MathSciNet review: 2085157