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

ed2k: ed2k stats

Год издания: 2007

Количество страниц: 518

Добавлена в каталог: 08.02.2014

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Предметный указатель

Chebyshev polynomial, dynamical properties      41
Chebyshev polynomial, even      41 329
Chebyshev polynomial, explicit formula      41 329
Chebyshev polynomial, fixed point      332 380
Chebyshev polynomial, fixed point, in characteristic       381
Chebyshev polynomial, in characteristic       380 381
Chebyshev polynomial, in characteristic two      381
Chebyshev polynomial, is monic of degree       329
Chebyshev polynomial, Julia set      30 41 336
Chebyshev polynomial, multiplier of fixed point      332 380
Chebyshev polynomial, multiplier of fixed point, in characteristic       381
Chebyshev polynomial, multiplier summation formula      380
Chebyshev polynomial, not conjugate to monomial      381
Chebyshev polynomial, odd      41 329
Chebyshev polynomial, periodic point      41
Chebyshev polynomial, recurrence relation      41 329
Chebyshev polynomial, twist      336
Chebyshev polynomial, uniform boundedness of periodic points      137
Chebyshev polynomial, value of T2, T3, T4      41
Chordal metric      102
Chordal metric, -adic      45 46 144 243
Chordal metric, complex      11 144
Chordal metric, dehomogenized form      11
Chordal metric, distance between closed disks      268
Chordal metric, effect of linear fractional transformation      76
Chordal metric, equals distance on Riemann sphere      11 35
Chordal metric, invariant maps for      35
Chordal metric, inversion is isometry      35
Chordal metric, Lipschitz property satisfied      11 24 36 56
Chordal metric, on inverse image of rational map      115 144
Chordal metric, related to Euclidean distance      119
Chordal metric, relation to cross-ratio      72
Chordal metric, resultant measures expansion      56 76
Chordal metric, value is between zero and one      11
Chordal sup norm      269
circle      242
Circle, is open and closed      243
Class field theory      202 204
Class number      350
Class number, of quadratic imaginary field      368
Closed branch of Berkovich disk      300
Closed disk      242
Closed disk, closed under addition      243
Closed disk, distance between      268
Closed disk, equivalent nested sequences      322
Closed disk, Gel’fond topology restricts to $\mathbb{C}_{p}$ topology      301 323
Closed disk, image of $\bar{D}(0,1)$ by $PGL_{2}$       317
Closed disk, image under inversion      321
Closed disk, in $\mathbb{P}^{1}$       243
Closed disk, is open      243
Closed disk, is ring of integers      243
Closed disk, nested sequence      295 322
Closed disk, radius of image by polynomial      321
Closed disk, rational radius      243 278 295
Closed disk, standard collection      277
cm      see complex multiplication
Coarse moduli space      160
Coboundary      201 202 209
Coboundary, trivial when extended      203
Cocycle      201 202
Cocycle corresponds to twist      203
Cocycle, $PGL_{2}$ gives twist of $\mathbb{P}^{1}$       211
Cocycle, cohomologous      202
Cocycle, determines field of moduli      209
Cohn’s theorem      443
Cohomology group      202
Cohomology group, maps to Brauer group      204
Cohomology group, twists map to      202 236
Cohomology set      202
Commutative algebraic group      375
Commutative algebraic group, universal cover      375
Commuting maps      8
Commuting maps, Chebyshev polynomial      332
Commuting maps, Fatou set      378
Commuting maps, have same canonical height      137
Commuting maps, have same preperiodic points      38
Commuting maps, higher-dimensional      378
Commuting maps, Julia set      378
Commuting maps, Latt $\grave{e}$ s      382
Commuting maps, monomial      325 326 380
Commuting maps, polynomial      378
Commuting maps, power map      326 380
Commuting rational map      378
Complete field      240
Complete field, $\mathbb{C}_{p}$       242
Complete field, circle      242
Complete field, closed disk      242
Complete field, extension      241
Complete field, holomorphic function      244
Complete field, open disk      242
Complete field, power series converges iff $a_{n}\rightarrow 0$       244
Complete field, projective space compact iff locally compact      243
Completely invariant set      17 258 266
Completely invariant set, boundary of Julia set      23
Completely invariant set, closed      267
Completely invariant set, component at infinity      40
Completely invariant set, contains Berkovich Julia set      311
Completely invariant set, Fatou set      23 40 266
Completely invariant set, finite      16 266
Completely invariant set, Julia set      23 40 266
Completion of a field      83
Completion of a field, valued      241
Complex chordal metric      11 144
Complex multiplication      32 341 348
Complex multiplication, by $\mathbb{Z}[i]$       341
Complex multiplication, by $\mathbb{Z}[\rho]$       341
Complex multiplication, degree of an endomorphism      349
Complex multiplication, fractional ideal gives curve with      350
Complex multiplication, ideal class group      350
Complex multiplication, is quadratic imaginary      349
Complex orbifold      176
Complex projective line      see Projective line
Complex torus      33
Complex torus, fundamental domain      33 127
Conformal map of Mandelbrot set complement      167
Conjugacy class in space of rational maps      174
Conjugation      11
Conjugation, canonical height is invariant      137
Conjugation, linear      11 173
Conjugation, new coordinate functions      218
Connectivity, Berkovich space      276
Connectivity, disk      276
Connectivity, Julia set      26 165
Connectivity, Mandelbrot set      167
Connectivity, rigid analytic      276
Continuity      22
Continuity, equi      254 264 265 271 306 313
Continuity, uniform      39 254 313
Continuous function      10 253
Continuous function, on metric space      22
Convex hull      249
Coordinate ring      see Affine coordinate ring
Counting Lemma      425
Critical point      12 284 353
Critical point, bounded orbit      26 165
Critical point, distinct modulo       237
Critical point, image under automorphism      234
Critical point, Julia set with no      285
Critical point, number of for rational map      14 37
Critical point, number of for separable map      37
Critical point, of Latt $\grave{e}$ s projection      384
Critical point, orbit for quadratic map      26 165
Critical point, periodic      38
Critical point, postcritical set      280
Critical point, ramification index      12 36
Critical point, rational map with two      233
Critical point, reduction is if point is attracting      78
Critical point, repelling      166
Critical point, strictly preperiodic      26 166
Critical point, tamely ramified      37

Critical point, totally ramified      12
Critical point, unbounded orbit      26
Critical point, whose reduction is periodic      78
Critical point, wildly ramified      37
Critical value      353 382
Critical value, distinct modulo       237
Cross-ratio      71 132 348
Cross-ratio, -invariant in terms of      348
Cross-ratio, permutation invariant      79
Cross-ratio, relation to chordal metric      72
Cubes, sum of two      141
Curve, cubic      30
Curve, function with given divisor      214
Curve, genus      15 37
Curve, of genus at least two      107
Curve, quotient by finite group      161
Curve, Riemann — Roch theorem      214
Curve, twist      200 211 215 236
Cuspidal cubic map      193
Cycle of -periodic points      150 226
Cyclic automorphism group      204
Cyclic group      197
Cyclotomic extension, maximal      95
Cyclotomic field      69 123
Cyclotomic field, dynamical analog of      123
Cyclotomic field, Frobenius      130
Cyclotomic field, Galois group      129
Cyclotomic polynomial      39 148 224
Cyclotomic polynomial, resultant with multiplier polynomial      226
Cyclotomic unit      69 72 133 328
Cyclotomic unit, action of Galois      133
Degenerate point for involution on K3 surface      415
Degenerate quadratic map      194
Degree two map induces involution      436 437
Degree, map on divisor class group      405
Degree, map on divisor group      405
Degree, map on function fields      44
Degree, map on Picard group      405
Degree, of algebraically stable morphism      396
Degree, of composition of affine morphisms      392
Degree, of divisor      339 405
Degree, of endomorphism      349
Degree, of family of morphisms      171
Degree, of flexible Latt $\grave{e}$ s map      357
Degree, of inverse of affine automorphism      390
Degree, of isogeny      340
Degree, of iterate of affine automorphism      390
Degree, of Lattes map      34
Degree, of morphism      389
Degree, of polynomial      389
Degree, of rational map      10 89 388 389
Degree, two map induces involution      411
Degree, universal map of given      171
Derivative, formal      12 37 313
Derivative, norm of      252 313
Derivative, of Chebyshev polynomial      381
Derivative, of Chebyshev polynomial, in characteristic two      381
Dessins d’enfant      217
Diagonal map      150 216 226
Differential equation characterizes Chebyshev polynomial      334
Differential form used to define multiplier      19
Dihedral group      197 217 234 327
Dihedral group, infinite      419 435
Dimension of family of K3 surfaces      412 438
Diophantine approximation      104
Diophantine approximation, Baker’s theorem      107
Diophantine approximation, Dirichlet’s theorem      104 141
Diophantine approximation, effective version      107
Diophantine approximation, Roth’s theorem      104 108
Diophantine equation      4 104 202
Diophantine equation, Pell      106
Diophantine equation, Thue’s theorem      105
Dirac measure      128
Dirichlet’s theorem on Diophantine approximation      104 141
Disconnected Julia set      26
Discrete dynamical system      1
Discrete periodic point      428
Discrete valuation      44
Discriminant      132 134 146 368
Discriminant, elliptic curve      30 337 343 370 385
Discriminant, minimal      221 342
Discriminant, of dynatomic polynomial      226 227
Disk      242
Disk component      277 317
Disk component, image may not be disk component      283
Disk component, is maximal disk connected set      317
Disk component, is open      317
Disk component, map on induced by rational map      283
Disk component, no wandering      284
Disk component, nonarchimedean has simple form      278
Disk component, of $(z^{2} - z)/p$       283
Disk component, open set is disjoint union of      277 317
Disk component, periodic      283
Disk component, preperiodic      283
Disk component, same as path-connected component over $\mathbb{C}$       277
Disk component, set of      283
Disk component, wandering      283
Disk connected set      276 317
Disk connected set, maximal is disk component      317
Disk domain, no wandering      284 285
Disk, Berkovich      295
Disk, closed      242
Disk, closed under addition      243
Disk, connected      317
Disk, distance between      268
Disk, image of unit disk by $PGL_{2}$       317
Disk, image under inversion      321
Disk, in $\mathbb{P}^{1}$       243
Disk, in topological space      277 317
Disk, nested sequence      295
Disk, open      242
Disk, polynomial sends to disk      312
Disk, power series sends to disk      252 305
Disk, radius of image by polynomial      321
Disk, rational radius      243 248 252 278 295
Disk, Siegel      28
Disk, standard collection      277
Disk, sup norm of function on      247
Distance, between closed disks      268
Distance, between images via linear fractional transformation      269 316
Distance, logarithmic      102
Division polynomial      362 383
Division polynomial, resultant      383
Divisor      338 403
Divisor class group      403
Divisor class group, degree map      405
Divisor class group, exact sequence      404
Divisor class group, height map on      408
Divisor class group, pullback homomorphism      405
Divisor class, ample      406
Divisor class, base locus      406
Divisor class, very ample      406
Divisor group      339 403
Divisor group, degree map      339 405
Divisor group, summation map      339
Divisor, action of involution      418 419 438
Divisor, ample      406
Divisor, associated to a rational function      339
Divisor, associated vector space       406
Divisor, base locus      406 408
Divisor, degree of      339 405
Divisor, effective      406
Divisor, effective, positivity of height      408
Divisor, group of principal      339
Divisor, intersection on K3 surface      434
Divisor, is difference of effective      406
Divisor, linear equivalence      404
Divisor, linearly equivalent give same height      408
Divisor, local ring      403
Divisor, of a rational function      320 403

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