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

Induced rational map      389
Inertia group      344
Infinite dihedral group      419
Integer point, counting on Markoff variety      437
Integer point, cutoff for in orbits      375
Integer point, in orbit      3 108 109 112 143 145
Integer point, in orbit of       8
Integer point, on $\mathbb{P}^{1}$ minus 3 points      106
Integer point, on elliptic curve      372
Integer point, on Markoff variety      437
Integer point, orbits with many      110 111 143 375
Integer point, uniform bound in orbits      112 372 385
Integer point, value of rational function      143
Integrable map      429
Integrable map, level curve is elliptic curve      430
Integrable map, level set      431
Integrable map, McMillan family      443
Integrable map, QRT family      443
Intersection theory      434
Invariant differential      344 359 367
Invariant differential, is holomorphic      345
Invariant differential, is translation-invariant      345
Invariant differential, transformation for       345
Invariant measure      307
Invariant set      16
Invariant set, completely      17 266
Invariant set, contains Berkovich Julia set      311
Invariant set, Fatou and Julia sets are completely      23
Invariant set, finite      16 266
Inverse function theorem      115 144
Inverse function theorem, over $\mathbb{C}_{p}$       312
Inverse of point on elliptic curve      337
Inversion maps disk toj      321
Involution      410
Involution, action on divisor      418 419 438
Involution, as linear transform on Picard group      420
Involution, composition is reversible affine automorphism      429
Involution, degenerate point      415
Involution, eigenvalue of      421
Involution, formulas to compute      433
Involution, indeterminacy locus      413
Involution, induced by degree 2 map      411 437
Involution, is morphism on most K3 surfaces      417
Involution, noncommuting      410
Involution, not a morphism      436
Involution, on Markoff variety      436
Involution, quotient by      410
Involution, relation between noncommuting      419
Involution, surface in $\mathbb{P}^{N}\time\mathbb{P}^{N}$ with two      436
Involution, three noncommuting      437
Irrational number approximated by rational number      104
Irrationally neutral periodic point      19
Isogenous elliptic curves      339
Isogeny      339
Isogeny, degree      340
Isogeny, dual      340 352
Isogeny, is homomorphism      340
Isogeny, is unramified      340 353
Isolated point      402
Isolated point, Julia set contains no      25
Isolated point, of bounded height      402
Isolated point, periodic      402
Isolated point, preperiodic      8
Isospectral family      186 188 356 362
Iterate, coordinate functions of      149
Iteration commutes with conjugation      11
Itinerary      258
Itinerary, as map on sequence space      260
Itinerary, left shift      260
Jacobian variety      369
Jointly regular affine morphisms      397 429
Jointly regular affine morphisms, height inequality      397
Julia set      22 255
Julia set, algebraic points      40
Julia set, all of $\mathbb{P}^{1}$       26 35 361
Julia set, backward orbit is dense      25
Julia set, Berkovich      306
Julia set, Berkovich, backward orbit dense in      311
Julia set, Berkovich, for bad reduction map      307 311
Julia set, Berkovich, for good reduction map      307 311
Julia set, Berkovich, has empty interior      311
Julia set, Berkovich, is connected or infinitely many components      311
Julia set, boundary is completely invariant      23
Julia set, boundary of Fatou set      25
Julia set, canonical height of points in      138
Julia set, chaotic behavior      22
Julia set, Chebyshev polynomial      30 41
Julia set, classical contained in Berkovich      306
Julia set, closure of repelling periodic points      41
Julia set, compact      275
Julia set, complement of Fatou set      22
Julia set, connected      26
Julia set, contained in closure of periodic points      273
Julia set, contains repelling periodic points      40 256
Julia set, disjoint from postcritical set closure      280
Julia set, effect of strictly preperiodic critical points      26
Julia set, equal to $\mathbb{Z}_{p}$       275
Julia set, equals closure of repelling periodic points      27 311
Julia set, filled      74 140
Julia set, Green function for filled      140
Julia set, has empty interior      267
Julia set, in two disks      257
Julia set, is closed      22
Julia set, is closure of backward orbit      267
Julia set, is completely invariant      23 74 266
Julia set, is empty for good reduction map      26 59 239
Julia set, is nonempty      25
Julia set, is perfect set      23 267 307 311
Julia set, is uncountable      267 307 311
Julia set, local canonical height of points in      141
Julia set, no critical points      285
Julia set, nonarchimedean      254
Julia set, nonempty interior      26
Julia set, of $(z^{2} -z)/p$       262 283
Julia set, of $z\cdot\cdot\cdot (z - d + 1 )/p$       315
Julia set, of $z^{2} - 2$       40
Julia set, of $z^{d}$       22
Julia set, of an iterate      24 255
Julia set, of Chebyshev polynomial      336
Julia set, of commuting maps      378
Julia set, of Latt $\grave{e}$ s map      35 361
Julia set, open set orbit omits at most one point      266
Julia set, orbit of open subset      27
Julia set, periodic points dense      23 315
Julia set, smallest closed completely invariant set      267
Julia set, strictly expanding map on      279 317
Julia set, support of canonical measure is      307
Julia set, topologically transitive map      263 315
Julia set, totally disconnected      23 26
Julia set, with no critical points      279
K3 surface      410
K3 surface, $H^{1}(S, \mathcal{O}_{S}) = 0$       412
K3 surface, $\mathcal{A}$ orbit      418
K3 surface, action of involution on divisors      418 438
K3 surface, arithmetic complexity of orbit      435
K3 surface, canonical height      421 423 435
K3 surface, canonical height, infinitely many points of bounded      435
K3 surface, canonical height, normalization conditions      422
K3 surface, canonical height, product $\hat{h}^{+}\hat{h}^{-}$       435
K3 surface, canonical height, transformation formulas      422
K3 surface, canonical height, zero iff finite orbit      423
K3 surface, degenerate point for involution      415
K3 surface, dimension of family of      412 438
K3 surface, divisors on      418
K3 surface, eigenvalue of involution on divisors      421 438
K3 surface, finitely many periodic points      435
K3 surface, finitely many points of bounded canonical height      423
K3 surface, Galois-invariant orbit      436
K3 surface, height functions on      420
K3 surface, height transformation rules      420

K3 surface, height zeta function      436
K3 surface, homogeneous forms associated to      412
K3 surface, in $\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$       437
K3 surface, indeterminacy locus of involution      413
K3 surface, intersection theory      434
K3 surface, involution      411
K3 surface, involution, as linear transform on Picard group      420
K3 surface, involution, example      411
K3 surface, involution, formulas      433
K3 surface, involution, is morphism on most      417 438
K3 surface, Kodaira dimension      412
K3 surface, nonsingular      438
K3 surface, number of points of bounded height      425
K3 surface, orbit is Zariski dense      435
K3 surface, Picard group of general      418
K3 surface, projections to $\mathbb{P}^{2}$       410
K3 surface, relation between noncommuting involutions      419
K3 surface, stabilizer of point on      425 434
K3 surface, with three involutions      437 438
Kawaguchi’s theorem      399
Kodaira dimension      412
Kronecker’s theorem      88 100
Krylov — Bogolubov theorem      127
Kummer sequence      205
L $\ddot{u}$ roth’s theorem      158
Lang’s height lower bound conjecture      101
Lang’s integer points conjecture      112
Latt $\grave{e}$ s diagram      365
Latt $\grave{e}$ s map      32 97 186 187 351
Latt $\grave{e}$ s map, affine minimal      372 385
Latt $\grave{e}$ s map, commuting      378
Latt $\grave{e}$ s map, composition of      382
Latt $\grave{e}$ s map, critical point of $\pi$       384
Latt $\grave{e}$ s map, critical values      353
Latt $\grave{e}$ s map, cutoff for integral points in orbit      375
Latt $\grave{e}$ s map, defining equation      32 351
Latt $\grave{e}$ s map, degree      34
Latt $\grave{e}$ s map, fixed points      34 366 382 384
Latt $\grave{e}$ s map, flexible      355
Latt $\grave{e}$ s map, for multiplication-by-      32
Latt $\grave{e}$ s map, for multiplication-by-2      32 351 370
Latt $\grave{e}$ s map, for multiplication-by-2, on curve      351 381
Latt $\grave{e}$ s map, for multiplication-by-2, on curve      351
Latt $\grave{e}$ s map, good reduction      362 383
Latt $\grave{e}$ s map, higher-dimensional      377
Latt $\grave{e}$ s map, in characteristic       366
Latt $\grave{e}$ s map, integral -invariant      370
Latt $\grave{e}$ s map, isospectral family      362
Latt $\grave{e}$ s map, iterate      382
Latt $\grave{e}$ s map, multiplier      34 186 358 366 382 384
Latt $\grave{e}$ s map, multiplier, at $\infty$       383
Latt $\grave{e}$ s map, multiplier, of fixed point      382
Latt $\grave{e}$ s map, multiplier, spectrum      362
Latt $\grave{e}$ s map, multiplier, summation formula      382
Latt $\grave{e}$ s map, nonconjugate      354
Latt $\grave{e}$ s map, nonconjugate, in characteristic 2      383
Latt $\grave{e}$ s map, nonconjugate, to polynomial      381
Latt $\grave{e}$ s map, padically -adically      361
Latt $\grave{e}$ s map, periodic points      186 358 366 382 384
Latt $\grave{e}$ s map, periodic points, all attracting      362
Latt $\grave{e}$ s map, periodic points, all nonrepelling      362
Latt $\grave{e}$ s map, periodic points, expanding      35
Latt $\grave{e}$ s map, periodic points, in Julia set      35 361
Latt $\grave{e}$ s map, postcritical set      353
Latt $\grave{e}$ s map, postcritically finite      353
Latt $\grave{e}$ s map, preperiodic points      32 352
Latt $\grave{e}$ s map, projection to $E/\Gamma$       365
Latt $\grave{e}$ s map, reduced diagram      365
Latt $\grave{e}$ s map, rigid      364
Latt $\grave{e}$ s map, set of is monoid      382
Latt $\grave{e}$ s map, torsion points map to preperiodic points      41
Latt $\grave{e}$ s map, translation fixed by $\Gamma$       365
Latt $\grave{e}$ s map, uniform bound for integral points in orbit      372 385
Latt $\grave{e}$ s map, uniform boundedness of periodic points      137
Lattice      33
Laurent series      245
Laurent series, coefficients uniquely determined by function      312
Laurent series, of Schwarzian derivative      232
Laurent series, product of      312
Least period      see exact period
Lefschetz principle      20 365
Left shift map      259 314
Left shift map, as itinerary map      260
Left shift map, backward orbit dense      314
Left shift map, backward orbit equidistributed      315
Left shift map, continuous      259
Left shift map, is topologically transitive      259
Left shift map, Lipschitz      259
Left shift map, periodic points      259
Left shift map, periodic points, are dense      314
Left shift map, properties of      259
Left shift map, topologically transitive      314
Left shift map, uniformly expanding      259
Lehmer’s conjecture      100 138
Lehmer’s conjecture, dynamical      101 138
Level curve of integrable map      430
Level set of an integrable map      431
Lie group      6
Lift      287
Lift, from $F_{p}$ to $\mathbb{Z}_{p}$       48
Lift, of affine morphism      389
Lift, of rational map      287 389
Line bundle, metrized      410
Line segment, from Gauss point to Type — IV point      322
Line segment, in Berkovich disk      298
Line segment, of Newton polygon      249
Linear conjugation      11 173
Linear conjugation, commutes with iteration      11
Linear equivalence      403
Linear equivalence, divisors give same height      408
Linear equivalence, pullback preserves      405
Linear fractional transformation      10
Linear fractional transformation, action of Galois      203
Linear fractional transformation, chordal sup norm      269
Linear fractional transformation, distance between images      269 316
Linear fractional transformation, effect on chordal metric      76
Linear fractional transformation, image of unit disk      317
Linear fractional transformation, Lipschitz constant      36
Linear fractional transformation, move three points      36
Linear fractional transformation, on $\mathbb{P}^{N}$       226
Linear fractional transformation, resultant of      76
Linear group, general      10
Linear group, projective special      175
Linear group, special      175
Lipschitz      11 24 36 56
Lipschitz, equi-      313
Lipschitz, holomorphic function      248 254 272
Lipschitz, shift map      259
Lipschitz, uniform family      39 254 264 265
Local canonical height      102 291
Local canonical height, associated to a divisor      320
Local canonical height, associated to an eigendivisor class      104
Local canonical height, computation of      103
Local canonical height, existence proof      291
Local canonical height, for a polynomial map      103 140 141
Local canonical height, good reduction case      103 141
Local canonical height, Green function and      291
Local canonical height, is harmonic      140
Local canonical height, Laplacian is invariant measure      104
Local canonical height, normalized      141
Local canonical height, of point in Fatou or Julia set      141
Local canonical height, properties of      102 291 320
Local canonical height, sum to global canonical height      103 293
Local canonical height, transformation law      320
Local degree      83
Local ring      67 79
Local ring, at a divisor      403
Local ring, normalized valuation      403
Locally compact      74 243 304
Locally compact, $\mathbb{C}_{p}$ is not      239 268 294 295

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