Silverman J. — The arithmetic of dynamical systems :: Электронная библиотека попечительского совета мехмата МГУ

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Silverman J. — The arithmetic of dynamical systems

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Название: The arithmetic of dynamical systems

Автор: Silverman J.

Аннотация:

This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs.This graduate-level text provides an entry for students into an active field of research and serves as a standard reference for researchers.

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Рубрика: Физика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2007

Количество страниц: 518

Добавлена в каталог: 08.02.2014

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Предметный указатель

Algebraic set      90
Algebraic set, attached to an ideal      90
Algebraic set, ideal of      90
Algebraic variety      89 147 159
Algebraic variety, dynamically affine      376
Algebraic variety, quotient by finite group      161
Algebraic variety, quotient by infinite group      161 174
Algebraically closed field      26 242
Algebraically closed field, $\mathbb{C}_{p}$ is an      239
Algebraically integrable map      430
Algebraically reversible affine automorphism      429
Algebraically stable morphism      396 433
Algebraically stable morphism, degree      396
Algebraically stable morphism, indeterminacy locus      396
Ample divisor      406
Ample divisor, bounded height is finite set      408
Analytic function      see Holomorphic function
Annulus      302
Archimedean absolute value      83
Arithmetic complexity, height in $\mathcal{M}_{d}$       221
Arithmetic complexity, minimal discriminant      221
Arithmetic complexity, of orbit on K3 surface      435
Arithmetic function      6
Arzel $\grave{a}$ — Ascoli theorem      25
Attracting basin of $\infty$       74 138 140 141
Attracting periodic point      19 47 326 362
Attracting periodic point, accumlates on a critical point      78
Attracting periodic point, basin of attraction      310
Attracting periodic point, every periodic point is an      78
Attracting periodic point, immediate basin of attraction      310
Attracting periodic point, in Berkovich space      310 324
Attracting periodic point, in Fatou set      22 40
Attracting periodic point, limit is repelling      274
Attracting periodic point, of Type I      310
Attracting periodic point, reduction is critical      78
Automorphism group      340 341
Automorphism group, $\{z^{\pm 1}\}$       205 234
Automorphism group, containing $\zeta_{n}z$       205 234
Automorphism group, contains $\mu_{2}$       235
Automorphism group, cyclic      197 204
Automorphism group, dihedral      217 234 327
Automorphism group, extension of cocycle to $PGL_{2}$       203
Automorphism group, involution      410
Automorphism group, is finite      196
Automorphism group, list of possible      197
Automorphism group, no twists if trivial      198
Automorphism group, noncommuting involutions      410 437
Automorphism group, of $z^{d}$       234
Automorphism group, of Chebyshev polynomial      332 336
Automorphism group, of Chebyshev polynomial, in characteristic       381
Automorphism group, of conjugate      196 234
Automorphism group, of elliptic curve      365
Automorphism group, of monomial map      327
Automorphism group, of monomial map, in characteristic       380
Automorphism group, of power map      327
Automorphism group, of power map, in characteristic       380
Automorphism group, of projective line      10
Automorphism group, order two      198 200
Automorphism group, points stabilized by      425 434
Automorphism group, polynomial map with nontrivial      234
Automorphism group, rational map      196
Automorphism group, symmetric group      197
Automorphism group, trivial for most maps      199 234
Automorphism, action on critical point      234
Automorphism, affine      390 427
Automorphism, affine regular      394
Automorphism, algebraically reversible      429
Automorphism, canonical height      431
Automorphism, finitely many periodic points      435
Automorphism, height inequality      399 431
Automorphism, height of periodic points      402
Automorphism, induced by degree two map      411 436 437
Automorphism, is subgroup of $PGL_{2}$       234
Automorphism, nondegenerate      226
Automorphism, of order two      96 197 235
Automorphism, rational      430
B $\acute{e}$ zivin’s theorem      274 275
B $\acute{e}$ zout’s theorem      428
Backward orbit, dense for shift map      314
Backward orbit, dense in Berkovich Julia set      311
Backward orbit, dense in Julia set      267
Backward orbit, equidistributed      315
Backward orbit, equidistributed for shift map      315
Backward orbit, of rational map      109 142
Bad reduction, becomes good reduction      218
Bad reduction, Berkovich Julia set of map with      307 311
Bad reduction, complex dynamics has      294
Bad reduction, composition of maps      77
Bad reduction, composition of maps can be good      77
Bad reduction, Green function of map with      294 318
Bad reduction, of $\phi(P)$ is not $\tilde{\phi}(\tilde{P})$       11
Bad reduction, product of primes of      221
Baire category theorem      268
Baker’s theorem      107 155
Banach algebra      298
Baragar’s theorem      443
Base locus      406 408
Basin of attraction      74 138 140 141 310
Basin of attraction, immediate      310 311
Benedetto’s theorem      284-286
Berkovich affine line      301
Berkovich affine line, as ringed space      304
Berkovich affine line, contained in Berkovich projective line      303
Berkovich affine line, contains $\mathbb{C}_{p}$       302 304 323
Berkovich affine line, Gel’fond topology restricts to $\mathbb{C}_{p}$ topology      304 323
Berkovich affine line, has no Gauss point      302
Berkovich affine line, is Hausdorff      304
Berkovich affine line, is locally compact      304
Berkovich affine line, is tree      302
Berkovich affine line, is uniquely path connected      304
Berkovich affine line, no Type-lV fixed points      321
Berkovich affine line, polynomial map      304
Berkovich disk      295
Berkovich disk, annulus      302
Berkovich disk, base of open sets      300
Berkovich disk, big model      323
Berkovich disk, branch      300
Berkovich disk, closed branch      300
Berkovich disk, closed of radius       297 301
Berkovich disk, closed unit      297
Berkovich disk, contains unit disk in $\mathbb{C}_{p}$       295 301 323
Berkovich disk, dynamics of $z^{2}$       307
Berkovich disk, Gauss point      297 301
Berkovich disk, Gel’fond topology      300 322
Berkovich disk, Gel’fond topology restricts to $\mathbb{C}_{p}$ topology      301 323
Berkovich disk, good reduction polynomial map      321 322
Berkovich disk, Hsia kernel      323
Berkovich disk, is compact      295 300 304
Berkovich disk, is connected      295
Berkovich disk, is Hausdorff      300 304
Berkovich disk, is metric space      295
Berkovich disk, is path-connected      300
Berkovich disk, is set of bounded seminorms      297
Berkovich disk, is set of seminorms      296
Berkovich disk, is uniquely path connected      304
Berkovich disk, Julia set of good reduction map      322
Berkovich disk, line segment from Gauss point to Type-lV point      322
Berkovich disk, line segment in      298
Berkovich disk, neutral fixed point      322
Berkovich disk, of radius       297 301
Berkovich disk, open branch      300
Berkovich disk, potential theory      323
Berkovich disk, radius of Type — IV point is positive      296
Berkovich disk, seminorm associated to each type of point      297
Berkovich disk, small model      323
Berkovich disk, topology on      299
Berkovich disk, union of annulus and open branch      302
Berkovich disk, visualizing      298
Berkovich Julia set, backward orbit dense in      311
Berkovich Julia set, equals closure of repelling periodic points      311

Berkovich Julia set, has empty interior      311
Berkovich Julia set, is connected or infinitely many components      311
Berkovich Julia set, is perfect set      307 311
Berkovich Julia set, is uncountable      307 311
Berkovich projective line      301 302
Berkovich projective line, as homogeneous two-variable seminorms      303 305
Berkovich projective line, as ringed space      304
Berkovich projective line, attracting fixed point      324
Berkovich projective line, basin of attraction      310
Berkovich projective line, branch at infinity      302
Berkovich projective line, canonical measure      306
Berkovich projective line, canonical measure, support of      307
Berkovich projective line, contains $\mathbb{P}^{1} (\mathbb{C}_{p})$       304 323
Berkovich projective line, contains Berkovich affine line      303
Berkovich projective line, domain of quasiperiodicity      310 311
Berkovich projective line, dynamics of $z^{2}$       307
Berkovich projective line, Fatou set      306 311
Berkovich projective line, Fatou set, contains classical      306
Berkovich projective line, Gel’fond topology restricts to $\mathbb{C}_{p}$ topology      304 323
Berkovich projective line, immediate basin of attraction      310 311
Berkovich projective line, is compact      304
Berkovich projective line, is Hausdorff      304
Berkovich projective line, is uniquely path connected      304
Berkovich projective line, Julia set      306
Berkovich projective line, Julia set, contains classical      306
Berkovich projective line, Julia set, equals closure of repelling periodic points      311
Berkovich projective line, Julia set, is nonempty      307
Berkovich projective line, Julia set, of bad reduction map      307 311
Berkovich projective line, Julia set, of good reduction map      307 311 322
Berkovich projective line, Montel theorem      310 311
Berkovich projective line, rational map      305
Berkovich projective line, recurrent point      310
Berkovich space      294
Berkovich space, attracting fixed point      324
Berkovich space, attracting periodic point      310
Berkovich space, basin of attraction      310
Berkovich space, canonical measure      6 306
Berkovich space, canonical measure, support of      307
Berkovich space, connectivity of      276
Berkovich space, different centers yield same point      295
Berkovich space, domain of quasiperiodicity      310 311
Berkovich space, dynamics of $z^{2}$       307
Berkovich space, dynamics on      304
Berkovich space, Fatou set      306 311
Berkovich space, fixed Gauss point      321
Berkovich space, Gauss point      297
Berkovich space, Gel’fond topology      322
Berkovich space, good reduction polynomial map      321 322
Berkovich space, Hsia kernel      323
Berkovich space, immediate basin of attraction      310 311
Berkovich space, invariant measure      104
Berkovich space, is set of seminorms      296
Berkovich space, Julia set      306
Berkovich space, Julia set, equals closure of repelling periodic points      311
Berkovich space, Julia set, is nonempty      307
Berkovich space, Julia set, of bad reduction map      307 311
Berkovich space, Julia set, of good reduction map      307 311 322
Berkovich space, Laplacian of local canonical height      104
Berkovich space, line segment from Gauss point to Type — IV point      322
Berkovich space, Montel theorem      310 311
Berkovich space, neutral fixed point      322
Berkovich space, path metric      306 323
Berkovich space, potential theory      323
Berkovich space, radius of a point      296
Berkovich space, radius of Type — IV point is positive      296
Berkovich space, recurrent point      310
Berkovich space, repelling periodic point      310
Berkovich space, rigid analytic space      298
Berkovich space, seminorm associated to each type of point      297
Berkovich space, topology on      299
Berkovich space, Type – I-IV points      295
Bicritical rational map      233
Bidegree      405 410
Bifurcation point      165
Bifurcation polynomial      165
Bifurcation polynomial, is a power      165 226 227
Big model of Berkovich disk      323
Big-Oh notation      93
Bihomogeneous polynomial      405 410
Binomial theorem      206 261
Birational map on Markoff variety      437
Bogomolov conjecture      129
Bogomolov conjecture, dynamical      129
Borel probability measure      127
Bounded height, finitely many points of      86 407
Bounded height, is finite set      408
Bounded height, Misiurewicz points have      167
Bounded seminorm on $\mathbb{C}_{p}[z]$       296
Bounded seminorm, automatically nonarchimedean      321
Bounded seminorm, properties      321
Bounded seminorm, set of is Berkovich disk      297
Branch at infinity in $\mathbb{P}^{\mathcal{B}}$       302
Branch of Berkovich disk      300
Brauer group      204
Brauer Siegel theorem      368
Brolin measure      307
Bulb      166
Canonical divisor      214
Canonical height      97 99 100 287
Canonical height, algorithm to compute      99 318
Canonical height, associated to an eigendivisor class      104
Canonical height, commuting maps have same      137
Canonical height, conjugation invariance      137
Canonical height, finitely many points of bounded      423
Canonical height, infinitely many points of bounded      435
Canonical height, is sum of Green functions      290 318
Canonical height, Lehmer’s conjecture      101 138
Canonical height, local      102 291 320
Canonical height, local for a polynomial map      140 141
Canonical height, lower bound for      101 137 138 221
Canonical height, normalization conditions      422
Canonical height, normalized local      141
Canonical height, of point in Fatou or Julia set      138
Canonical height, on abelian variety      409
Canonical height, on elliptic curve      409
Canonical height, on K3 surface      421 423 426 435 436
Canonical height, on K3 surface, with three involutions      438
Canonical height, product $\hat{h}^{+}\hat{h}^{-}$       435
Canonical height, quadratic map      137 138
Canonical height, regular affine automorphism      431
Canonical height, sum of local canonical heights      103 293
Canonical height, Tate construction      97
Canonical height, transformation formulas      422 432
Canonical height, uniqueness      98
Canonical height, zero iff finite orbit      423 431
Canonical height, zero iff preperiodic point      99
Canonical height, zeta function      436
Canonical measure      127 306
Canonical measure, on Berkovich space      306
Canonical measure, support is Julia set      307
Canonical measure, supported at Gauss point      307
Cardiod      166
Cauchy estimate for norm of derivative      252 313
Cauchy residue theorem      20 248 314
Cauchy sequence      98 240 296
Chain rule      18 360
Chain rule, calculate multiplier with      19 47
Change of variables      11
Chaos      3 22
Chebyshev polynomial      29 30 95 329
Chebyshev polynomial, algebraic properties      41 329
Chebyshev polynomial, alternative normalization      30 331
Chebyshev polynomial, automorphism group      332 336
Chebyshev polynomial, automorphism group, in characteristic       381
Chebyshev polynomial, binomial coefficient identity      380
Chebyshev polynomial, commutativity      41 329 332 378
Chebyshev polynomial, defining equation      30
Chebyshev polynomial, derivative identity      381
Chebyshev polynomial, derivative in characteristic      2 381
Chebyshev polynomial, differential equation characterizing      334

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