Book Review
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MathSciNet review:
2798321
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Book Information:
Author:
Joseph H. Silverman
Title:
The arithmetic of dynamical systems
Additional book information:
Graduate Texts in Mathematics, 241, Springer,
New York,
2007,
x+511 pp.,
ISBN 0-978-387-69903-5,
US$49.95$,
hardcover
Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
Robert L. Benedetto, Reduction, dynamics, and Julia sets of rational functions, J. Number Theory 86 (2001), no. 2, 175–195. MR 1813109, DOI 10.1006/jnth.2000.2577
Jean-Paul Bézivin, Sur les points périodiques des applications rationnelles en dynamique ultramétrique, Acta Arith. 100 (2001), no. 1, 63–74 (French). MR 1864626, DOI 10.4064/aa100-1-5
Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
Liang-Chung Hsia, Closure of periodic points over a non-Archimedean field, J. London Math. Soc. (2) 62 (2000), no. 3, 685–700. MR 1794277, DOI 10.1112/S0024610700001447
B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287
Loïc Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), no. 1-3, 437–449 (French). MR 1369424, DOI 10.1007/s002220050059
John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
Patrick Morton, Arithmetic properties of periodic points of quadratic maps. II, Acta Arith. 87 (1998), no. 2, 89–102. MR 1665198, DOI 10.4064/aa-87-2-89-102
Patrick Morton and Joseph H. Silverman, Rational periodic points of rational functions, Internat. Math. Res. Notices 2 (1994), 97–110. MR 1264933, DOI 10.1155/S1073792894000127
D. G. Northcott, Periodic points on an algebraic variety, Ann. of Math. (2) 51 (1950), 167–177. MR 34607, DOI 10.2307/1969504
Bjorn Poonen, The classification of rational preperiodic points of quadratic polynomials over $\textbf {Q}$: a refined conjecture, Math. Z. 228 (1998), no. 1, 11–29. MR 1617987, DOI 10.1007/PL00004405
Juan Rivera-Letelier, Dynamique des fonctions rationnelles sur des corps locaux, Astérisque 287 (2003), xv, 147–230 (French, with English and French summaries). Geometric methods in dynamics. II. MR 2040006
14. R. Rumely and M. Baker,
Analysis and dynamics on the Berkovich projective line,
preprint, math.NT/0407433, 2004.
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8
- 1.
- A. Beardon,
Iteration of Rational Functions,
Springer-Verlag, New York, 1991. MR 1128089
- 2.
- R. Benedetto,
Reduction, dynamics, and Julia sets of rational functions,
J. Number Theory 86 (2001), 175-195. MR 1813109
- 3.
- J.-P. Bézivin,
Sur les points périodiques des applications rationnelles en analyse ultramétrique,
Acta Arith. 100 (2001), 63-74. MR 1864626
- 4.
- L. Carleson and T. Gamelin,
Complex Dynamics,
Springer-Verlag, New York, 1991. MR 1230383
- 5.
- L.-C. Hsia,
Closure of periodic points over a non-archimedean field,
J. London Math. Soc. (2) 62 (2000), 685-700. MR 1794277
- 6.
- B. Mazur,
Modular curves and the Eisenstein ideal,
Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33-186. MR 0488287
- 7.
- L. Merel,
Bournes pour la torsion des courbes elliptiques sur les corps de nombres,
Invent. Math. 124 (1996), 437-449. MR 1369424
- 8.
- J. Milnor,
Dynamics in One Complex Variable: Introductory Lectures, 2nd ed.,
Vieweg, Braunschweig, 2000. MR 1721240
- 9.
- P. Morton,
Arithmetic properties of periodic points of quadratic maps, II,
Acta Arith. 87 (1998), 89-102. MR 1665198
- 10.
- P. Morton and J. Silverman,
Rational periodic points of rational functions,
Inter. Math. Res. Notices 2 (1994), 97-110. MR 1264933
- 11.
- D. Northcott,
Periodic points of an algebraic variety,
Ann. Math. 51 (1950), 167-177. MR 0034607
- 12.
- B. Poonen,
The classification of rational preperiodic points of quadratic polynomials over 
: a refined conjecture,
Math. Z. 228 (1998), 11-29. MR 1617987
- 13.
- J. Rivera-Letelier,
Dynamique des fonctions rationnelles sur des corps locaux,
Astérisque 287 (2003), 147-230. MR 2040006
- 14.
- R. Rumely and M. Baker,
Analysis and dynamics on the Berkovich projective line,
preprint, math.NT/0407433, 2004.
- 15.
- J. Silverman,
The Arithmetic of Elliptic Curves,
Springer-Verlag, New York, 1986. MR 0817210
- 16.
- J. Silverman,
Advanced Topics in the Arithmetic of Elliptic Curves,
Springer-Verlag, New York, 1994. MR 1312368
Review Information:
Reviewer:
Robert L. Benedetto
Affiliation:
Department of Mathematics and Computer Science, Amherst College, Amherst, Massachusetts 01002
Email:
rlb@cs.amherst.edu
Journal:
Bull. Amer. Math. Soc.
46 (2009), 157-164
DOI:
https://doi.org/10.1090/S0273-0979-08-01216-0
Published electronically:
August 29, 2008
Additional Notes:
The reviewer gratefully acknowledges the support of NSF Grant DMS-0600878.
Review copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
