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Book Information:
Authors:
Andrei Yu. Khrennikov and
Marcus Nilsson
Title:
p -Adic deterministic and random dynamics
Additional book information:
Kluwer Academic,
Dordrecht-Boston-London,
2004,
270 pp.,
ISBN 1-4020-2659-5,
US$99.00$
David K. Arrowsmith and Franco Vivaldi, Geometry of $p$-adic Siegel discs, Phys. D 71 (1994), no. 1-2, 222–236. MR 1264116, DOI 10.1016/0167-2789(94)90191-0
bakerrumely M. Baker and R. Rumely, Analysis and dynamics on the Berkovich projective line, arXiv:math.NT/0407433, 2004.
Robert L. Benedetto, Components and periodic points in non-Archimedean dynamics, Proc. London Math. Soc. (3) 84 (2002), no. 1, 231–256. MR 1863402, DOI 10.1112/plms/84.1.231
Amnon Besser and Christopher Deninger, $p$-adic Mahler measures, J. Reine Angew. Math. 517 (1999), 19–50. MR 1728549, DOI 10.1515/crll.1999.093
D. Bosio and F. Vivaldi, Round-off errors and $p$-adic numbers, Nonlinearity 13 (2000), no. 1, 309–322. MR 1734635, DOI 10.1088/0951-7715/13/1/315
John Bryk and Cesar E. Silva, Measurable dynamics of simple $p$-adic polynomials, Amer. Math. Monthly 112 (2005), no. 3, 212–232. MR 2125384, DOI 10.2307/30037439
Lee Brekke, Peter G. O. Freund, Mark Olson, and Edward Witten, Non-Archimedean string dynamics, Nuclear Phys. B 302 (1988), no. 3, 365–402. MR 947888, DOI 10.1016/0550-3213(88)90207-6
Gregory S. Call and Joseph H. Silverman, Canonical heights on varieties with morphisms, Compositio Math. 89 (1993), no. 2, 163–205. MR 1255693
Zaqueu Coelho and William Parry, Ergodicity of $p$-adic multiplications and the distribution of Fibonacci numbers, Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2, vol. 202, Amer. Math. Soc., Providence, RI, 2001, pp. 51–70. MR 1819181, DOI 10.1090/trans2/202/06
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
Peter G. O. Freund and Mark Olson, Non-Archimedean strings, Phys. Lett. B 199 (1987), no. 2, 186–190. MR 919703, DOI 10.1016/0370-2693(87)91356-6
Peter G. O. Freund and Edward Witten, Adelic string amplitudes, Phys. Lett. B 199 (1987), no. 2, 191–194. MR 919704, DOI 10.1016/0370-2693(87)91357-8
M. Herman and J.-C. Yoccoz, Generalizations of some theorems of small divisors to non-Archimedean fields, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 408–447. MR 730280, DOI 10.1007/BFb0061427
Liang-Chung Hsia, A weak Néron model with applications to $p$-adic dynamical systems, Compositio Math. 100 (1996), no. 3, 277–304. MR 1387667
Andrei Khrennikov, Non-Archimedean analysis: quantum paradoxes, dynamical systems and biological models, Mathematics and its Applications, vol. 427, Kluwer Academic Publishers, Dordrecht, 1997. MR 1746953, DOI 10.1007/978-94-009-1483-4
Andrei Khrennikov, Shinichi Yamada, and Arnoud van Rooij, The measure-theoretical approach to $p$-adic probability theory, Ann. Math. Blaise Pascal 6 (1999), no. 1, 21–32. MR 1693138
Hua-Chieh Li, $p$-adic dynamical systems and formal groups, Compositio Math. 104 (1996), no. 1, 41–54. MR 1420709
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
D. A. Lind, Ergodic group automorphisms are exponentially recurrent, Israel J. Math. 41 (1982), no. 4, 313–320. MR 657863, DOI 10.1007/BF02760537
Douglas Lind and Klaus Schmidt, Bernoullicity of solenoidal automorphisms and global fields, Israel J. Math. 87 (1994), no. 1-3, 33–35. MR 1286813, DOI 10.1007/BF02772981
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
D. W. Masser, Mixing and linear equations over groups in positive characteristic, Israel J. Math. 142 (2004), 189–204. MR 2085715, DOI 10.1007/BF02771532
Patrick Morton and Joseph H. Silverman, Periodic points, multiplicities, and dynamical units, J. Reine Angew. Math. 461 (1995), 81–122. MR 1324210, DOI 10.1515/crll.1995.461.81
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
Klaus Schmidt, Dynamical systems of algebraic origin, Progress in Mathematics, vol. 128, Birkhäuser Verlag, Basel, 1995. MR 1345152
silverman J. Silverman, www.math.brown.edu/$\tilde {\ }$jhs/MA0272/ArithDynRefsOnly.pdf.
vivaldi F. Vivaldi, www.maths.qmw.ac.uk/$\tilde {\ }$fv/database/algdyn.bib.
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
Thomas B. Ward, Additive cellular automata and volume growth, Entropy 2 (2000), no. 3, 142–167. MR 1882488, DOI 10.3390/e2030142
S. A. Juzvinskiĭ, Calculation of the entropy of a group-endomorphism, Sibirsk. Mat. . 8 (1967), 230–239 (Russian). MR 0214726
MR1264116 D. Arrowsmith and F. Vivaldi, Geometry of $p$-adic Siegel discs, Phys. D 71 (1994), no. 1-2, 222–236.
bakerrumely M. Baker and R. Rumely, Analysis and dynamics on the Berkovich projective line, arXiv:math.NT/0407433, 2004.
MR1863402 R.L. Benedetto, Components and periodic points in non-Archimedean dynamics, Proc. London Math. Soc. (3) 84 (2002), no. 1, 231–256.
MR1728549 A. Besser and C. Deninger, $p$-adic Mahler measures, J. Reine Angew. Math. 517 (1999), 19–50.
MR1734635 D. Bosio and F. Vivaldi, Round-off errors and $p$-adic numbers, Nonlinearity 13 (2000), no. 1, 309–322.
MR2125384 J. Bryk and C. E. Silva, Measurable dynamics of simple $p$-adic polynomials, Amer. Math. Monthly 112 (2005), no. 3, 212–232.
MR947888 L. Brekke, P. Freund, M. Olson, and E. Witten, Non-Archimedean string dynamics, Nuclear Phys. B 302 (1988), no. 3, 365–402.
MR1255693 G. S. Call and J. H. Silverman, Canonical heights on varieties with morphisms, Compositio Math. 89 (1993), no. 2, 163–205.
MR1819181 Z. Coelho and W. Parry, Ergodicity of $p$-adic multiplications and the distribution of Fibonacci numbers, Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2, vol. 202, Amer. Math. Soc., Providence, RI, 2001, pp. 51–70.
MR2031042 M. Einsiedler and D. Lind, Algebraic $\mathbb Z^ d$-actions of entropy rank one, Trans. Amer. Math. Soc. 356 (2004), no. 5, 1799–1831 (electronic).
MR919703 P. Freund and M. Olson, Non-Archimedean strings, Phys. Lett. B 199 (1987), no. 2, 186–190.
MR919704 P. Freund and E. Witten, Adelic string amplitudes, Phys. Lett. B 199 (1987), no. 2, 191–194.
MR730280 M. Herman and J.-C. Yoccoz, Generalizations of some theorems of small divisors to non-Archimedean fields, Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 408–447.
MR1387667 L.-C. Hsia, A weak Néron model with applications to $p$-adic dynamical systems, Compositio Math. 100 (1996), no. 3, 277–304.
MR1746953 A. Khrennikov, Non-Archimedean analysis: quantum paradoxes, dynamical systems and biological models, Mathematics and its Applications, vol. 427, Kluwer Academic Publishers, Dordrecht, 1997.
MR1693138 A. Khrennikov, S. Yamada, and A. van Rooij, The measure-theoretical approach to $p$-adic probability theory, Ann. Math. Blaise Pascal 6 (1999), no. 1, 21–32.
MR1420709 H.-C. Li, $p$-adic dynamical systems and formal groups, Compositio Math. 104 (1996), no. 1, 41–54.
MR0460593 D. A. Lind, The structure of skew products with ergodic group automorphisms, Israel J. Math. 28 (1977), no. 3, 205–248.
MR657863 —, Ergodic group automorphisms are exponentially recurrent, Israel J. Math. 41 (1982), no. 4, 313–320.
MR1286813 D. Lind and K. Schmidt, Bernoullicity of solenoidal automorphisms and global fields, Israel J. Math. 87 (1994), no. 1-3, 33–35.
MR92j:22013 D. Lind, K. Schmidt, and T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math. 101 (1990), no. 3, 593–629.
MR961739 D. A. Lind and T. Ward, Automorphisms of solenoids and $p$-adic entropy, Ergodic Theory Dynam. Systems 8 (1988), no. 3, 411–419.
MR2085715 D. W. Masser, Mixing and linear equations over groups in positive characteristic, Israel J. Math. 142 (2004), 189–204.
MR1324210 P. Morton and J.H. Silverman, Periodic points, multiplicities, and dynamical units, J. Reine Angew. Math. 461 (1995), 81–122.
MR2040006 J. Rivera-Letelier, Dynamique des fonctions rationnelles sur des corps locaux, Astérisque (2003), no. 287, xv, 147–230.
MR97c:28041 K. Schmidt, Dynamical systems of algebraic origin, Birkhäuser Verlag, Basel, 1995.
silverman J. Silverman, www.math.brown.edu/$\tilde {\ }$jhs/MA0272/ArithDynRefsOnly.pdf.
vivaldi F. Vivaldi, www.maths.qmw.ac.uk/$\tilde {\ }$fv/database/algdyn.bib.
MR1619569 T. Ward, Almost all $S$-integer dynamical systems have many periodic points, Ergodic Theory Dynam. Systems 18 (1998), no. 2, 471–486.
MR1882488 —, Additive cellular automata and volume growth, Entropy 2 (2000), no. 3, 142–167 (electronic).
MR0214726 S. A. Yuzvinskiĭ. Calculation of the entropy of a group-endomorphism, Sibirsk. Mat. Z̆., 8 (1967), 230–239.
Review Information:
Reviewer:
Thomas Ward
Affiliation:
University of East Anglia
Email:
t.ward@uea.ac.uk
Journal:
Bull. Amer. Math. Soc.
43 (2006), 133-137
Published electronically:
October 7, 2005
Review copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
