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

Rational map, number of critical points      14 37
Rational map, number of nonrepelling periodic cycles      27
Rational map, number of periodic points      39
Rational map, of polynomial type      143
Rational map, omitting three points      25
Rational map, on Berkovich space      305
Rational map, orbit of collection of      143
Rational map, orbit with many integral points      110 111 143 375
Rational map, parabolic component      28
Rational map, period polynomial      149
Rational map, periodic point      394
Rational map, periodic point, field generated by      122
Rational map, periodic point, index summation formula      20 38 380
Rational map, periodic subvariety      127
Rational map, polynomial if has totally ramified fixed point      17
Rational map, postcritical set      353
Rational map, power map      29 321 325
Rational map, power map, in characteristic       380
Rational map, preperiodic subvariety      127 128
Rational map, primitive periodic point      122
Rational map, product of primes of bad reduction      221
Rational map, pullback of divisor      404
Rational map, pullback of divisor, linearly equivalent      405
Rational map, pushforward of divisor      404
Rational map, quadratic is polynomial iff $\sigma_{1}(\phi) = 2$       232
Rational map, quadratic twist      200
Rational map, ramification index      12 36
Rational map, ramification index, of iterate      117
Rational map, ramification point      see critical point
Rational map, recurrent point      284 310
Rational map, reducible dynatomic polynomial      235
Rational map, reduction modulo a prime      52
Rational map, repelling periodic point in Julia set      256
Rational map, resultant      56 58 75 112 372
Rational map, resultant, of composition      77
Rational map, resultant, of conjugate      76
Rational map, resultant, of iterate      184
Rational map, Riemann — Hurwitz formula      13 217
Rational map, self similarity      196
Rational map, semistable      179
Rational map, separable      37
Rational map, separable reduction      64
Rational map, set of ( $Rat_{d}$ )      169
Rational map, Siegel disk      28
Rational map, space of conjugacy classes of      174
Rational map, stable      179
Rational map, strictly expanding on Julia set      279 317
Rational map, symmetric polynomial of multipliers      183 187 232 367
Rational map, Taylor series      190
Rational map, totally ramified fixed point      37
Rational map, totally ramified point      12
Rational map, twist      197
Rational map, twist, same image in $\mathcal{M}_{d}$       198
Rational map, twist, with good reduction outside       237
Rational map, uniform bound for integral points in orbit      112 372 385
Rational map, uniformly continuous family      254 313
Rational map, uniformly Lipschitz      254 256 313
Rational map, universal of degree       171
Rational map, valuation of minimal resultant      219
Rational map, very highly ramified      231 284
Rational map, wandering disk domains exist over $\mathbb{C}_{p}$       285
Rational map, Weierstrass class      223 237
Rational map, wildly critical point      284
Rational map, with $Aut(\phi) = \mathcal{S}_{3}$       234
Rational map, with $\zeta_{n}z$ in $Aut(\phi)$       205 234
Rational map, with same minimal discriminant      221
Rational map, with two critical points      233
Rational numbers      7
Rational numbers, absolute values on      44 312
Rational radius      243 252 295 317
Rational variety      232
Rationally neutral periodic point      19 47
Rationally neutral periodic point, in hyperbolic component      28
Raynaud’s theorem      127
Real fixed point      427
Realizable sequence      6
Recurrent point      3 284 310
Recurrent point, critical in Fatou set is periodic      284 317
Recurrent point, critical in Julia set      284
Reduction, good      see good reduction
Reduction, modulo on elliptic curve      342
Reduction, of a point modulo a prime      48
Reduction, of a rational map      52
Reduction, of periodic point      61 62 66 78
Reduction, of torsion on elliptic curve      62 343 364
Reduction, relation to -adic distance      49
Reduction, respected by fraction linear transformation      50
Reduction, separable      64
Regular affine automorphism      391 394 427
Regular affine automorphism, algebraically stable      396
Regular affine automorphism, canonical height      431
Regular affine automorphism, canonical height, zero iff finite orbit      431
Regular affine automorphism, degree      394
Regular affine automorphism, degree, of inverse      394 395
Regular affine automorphism, height inequality      399 431
Regular affine automorphism, height of wandering point      432
Regular affine automorphism, indeterminacy locus      394
Regular affine automorphism, iterate is regular      394
Regular affine automorphism, number of periodic points      428
Regular affine automorphism, orbit-counting function      432
Regular affine automorphism, periodic points      394 428
Regular function on $Rat_{d}$       169 174
Repelling periodic point      19 326 361
Repelling periodic point, closure equals Julia set      274 311
Repelling periodic point, dense in Julia set      23 263 315
Repelling periodic point, in Berkovich space      310
Repelling periodic point, in Julia set      22 256
Repelling periodic point, limit of attracting      274
Repelling periodic point, of Type II      310
Repelling periodic point, one implies infinitely many      274
Representation of Galois on elliptic curve torsion      344
Residual degree      310
Residue fixed-point index      38
Residue formula      314
Residue, abstract      313
Residue, at simple pole      313
Residue, of a rational function      313
Residue, rational function has finitely many nonzero      314
Resultant      53 107
Resultant, chordal metric formula      76
Resultant, determinant formula      54
Resultant, homogeneity of      54 219
Resultant, measures expansion      56 76
Resultant, minimal      219
Resultant, nonvanishing defines $Rat_{d}$       169
Resultant, normalized      77
Resultant, of a composition of maps      77
Resultant, of a linear fractional transformation      76
Resultant, of a rational map      56 58 75 112 218 372
Resultant, of conjugate      76
Resultant, of coordinate functions of iterate      227
Resultant, of division polynomial      383
Resultant, of dynatomic polynomial      165
Resultant, of iterate      184
Resultant, of quadratic polynomial      75
Resultant, product of differences of roots      53 75
Resultant, properties of      53
Resultant, transformation formulas      75 173 219
Resultant, unit iff good reduction      58 218
Resultant, with a linear polynomial      75
Reversibility      429
Reversible affine automorphism      429
Reversible affine automorphism, composition of involutions      429
Reversible affine automorphism, H $\acute{e}$ non map      430
Reversible affine automorphism, number of orbits over a finite field      430
Riemann sphere      see projective line
Riemann surface      37
Riemann surface, formula      37
Riemann surface, Riemann — Hurwitz formula      37
Riemann — Hurwitz formula      13 65 109 117 340 347 379

Riemann — Hurwitz formula, application to finite invariant set      16
Riemann — Hurwitz formula, for curves      15 37
Riemann — Hurwitz formula, is local global formula      15
Riemann — Hurwitz formula, weak version      15
Riemann — Roch theorem      214 337
Rigid analytic connectivity      276
Rigid analytic space      298
Rigid Latt $\grave{e}$ s map, fixed points      366
Rigid Latt $\grave{e}$ s map, good reduction      383
Rigid Latt $\grave{e}$ s map, periodic points      366
Ring of -integers      83
Ring of integers      83
Ritt’s theorem      378
Rivera — Letelier, classification theorem      310
Rivera — Letelier, Fatou domain classification      240
Rivera — Letelier, strong Montel theorem      311
Root of holomorphic function determined by Newton polygon      249
Root of hyperbolic component of Mandelbrot set      166
Root of unity, iff height one      88 100
Root of unity, multiplier      151 226
Root of unity, primitive      148
Roth’s theorem      104 106 120
Roth’s theorem, is ineffective      107
Roth’s theorem, over number fields      108
Schanual’s theorem      135
Scheme      59 303
Scheme, good reduction      59 61 77
Scheme, moduli      179 186 189
Scheme, multiplicative group      29
Schwarzian derivative      231
Segre embedding      406
Self similarity      196
Seminorm, action of polynomial map      304
Seminorm, action of rational map      305
Seminorm, associated to each Berkovich point type      297
Seminorm, associated to nested sequence of disks      297 322
Seminorm, is norm for Type — II,-III,-IV points      297 321
Seminorm, on $\mathbb{C}_{p}[z]$       296
Seminorm, properties      321
Seminorm, set of bounded is Berkovich disk      297
Seminorm, two variable homogeneous      303 305
Semistable      178
Semistable, locus of $PSL_{2}$ action on $\mathbb{P}^{2d+1}$       178
Semistable, numerical criterion      178
Semistable, quotient      179
Semistable, rational map      179
Separable field extension      37
Separable rational map      37
Separable rational map, number of critical points      37
Sequence space      258 314
Sequence space, backward orbit dense      314
Sequence space, backward orbit equidistributed      315
Sequence space, left shift      259 314
Sequence space, metric on      258 314
Sequence space, shift map periodic points dense      314
Sequence space, shift map topologically transitive      314
Shafarevich conjecture      442
Sheshadri’s theorem      179
Shift map      259 314
Shift map, as itinerary map      260
Shift map, backward orbit dense      314
Shift map, backward orbit equidistributed      315
Shift map, continuous      259
Shift map, is topologically transitive      259
Shift map, Lipschitz      259
Shift map, periodic points      259
Shift map, periodic points dense      314
Shift map, properties of      259
Shift map, topologically transitive      314
Shift map, uniformly expanding      259
Siegel disk      28
Siegel’s theorem, integer points on $\mathbb{P}^{1}$ minus 3 points      106 110
Siegel’s theorem, integrality of points on elliptic curves      107
Siegel’s theorem, is ineffective      107
Siegel’s theorem, over number fields      108
Simple algebraic group      377
Small model of Berkovich disk      323
Solenoid      6
Special fiber      77
Special linear group      175
Sphere formula      13
Spherical measure      36
Spider algorithm      168
Stabilizer of point on K3 surface      425 434
Stable locus      176
Stable locus, of $PSL_{2}$ action on $\mathbb{P}^{2d+1}$       178
Stable numerical criterion      178
Stable quotient      179
Stable rational map      179
Standard collection of disks      277
Strange Attractor      391
Sullivan no wandering domains theorem      28
Sum of two cubes      141
Sum of two squares      200 236
Sup norm, chordal      269
Sup norm, Gauss norm is      296
Sup norm, of a holomorphic function      247
Sup norm, of a point      90 288
Sup norm, of a polynomial      91 288
Superattracting periodic point      19
Supersingular elliptic curve      378
Support of a divisor      403
Surface, involution induced by degree 2 map      436
Surface, K3      410
Surface, Kodaira dimension      412
Symbolic dynamics      258 314
Symbolic dynamics, backward orbit dense      314
Symbolic dynamics, backward orbit equidistributed      315
Symbolic dynamics, shift map periodic points dense      314
Symbolic dynamics, shift map topologically transitive      314
Symmetric group      124
Symmetric group, rational map with $Aut(\phi) = \mathcal{S}_{3}$       234
Symmetric group, wreath product with cyclic group      125
Symmetric polynomial of multipliers      183 187 232 367
Symmetric polynomial, elementary      180
Symmetric polynomial, of multipliers      180
Szpiro conjecture      221
Szpiro — Tucker theorem      442
Tamely ramified critical point      37
Tate algebra      297
Tate algebra, Gauss norm      298
Tate construction of canonical height      97
Tate module      344
Tate module, -adic representation      344
Tate’s algorithm      342
Taylor series      12 63 67 190
Telescoping sum      98
Tetrahedral group      197
Thue — Mahler theorem      108
Thue’s theorem      105
Thue’s theorem, is ineffective      107
Thue’s theorem, over number fields      108
Topological equicontinuity      306
Topological group      8
Topological space, disk component      277 317
Topological space, disk connected set      317
Topological space, dynamical system on      3
Topological space, equicontinuity      306
Topological space, generalized disk      277 317
Topological space, path connected      304
Topological space, uniquely path connected      304
Topological wandering point      108
Topologically transitive map      259 263 315
Topologically transitive map, shift is      314
topology      6
Topology, on Berkovich disk      299
Topology, profinite      211
Torsion point, elliptic curve      41
Torsion point, field generated by on elliptic curve      344
Torsion point, of abelian group is preperiodic      2 41

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