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

Locally constant function      318
Locus of indeterminacy      389
Locus of indeterminacy, affine automorphism      392
Locus of indeterminacy, affine morphism      392
Locus of indeterminacy, algebraically stable morphism      396
Locus of indeterminacy, jointly regular affine morphisms      397
Logarithmic distance      102
Logarithmic height      93
Logarithmic height, absolute      93
Logarithmic singularity      140
Long Exact Sequence      205
Lyubich measure      307
M $\ddot{o}$ bius $\mu$ function      39 148 149 164 224
M $\ddot{o}$ bius inversion formula      148 224
M $\ddot{o}$ bius transformation      see linear fractional transformation
Mahler measure      138
Mandelbrot set      26 165
Mandelbrot set, analytic description of Misiurewicz points      229
Mandelbrot set, bulb      166
Mandelbrot set, complement conformal to unit disk exterior      167 168
Mandelbrot set, component of interior      166
Mandelbrot set, in disk of radius 2      166
Mandelbrot set, is connected      167
Mandelbrot set, Misiurewicz point      166
Mandelbrot set, Misiurewicz point, analytic description      168
Mandelbrot set, Misiurewicz point, dense in      166
Mandelbrot set, Misiurewicz point, in boundary      166
Mandelbrot set, root of hyperbolic component      166
Mandelbrot set, spider algorithm      168
Mandelbrot set, uniformization      167 168
Manes’s theorem      235
Manin — Mumford conjecture      127
Manin — Mumford conjecture, dynamical      127
Markoff equation      436
Markoff equation, integer points      437
Markoff equation, Markoff triple      437
Markoff equation, unicity conjecture      437
Markoff equation, variety is rational      437
Masser Oesterl $\acute{e}$ conjecture      372 373
Maximum Modulus Principle      248 250 252 253 268 296 298 305 321
Maximum modulus principle, false if not algebraically closed      252
Maximum modulus principle, for rational functions      250
Mazur Kamienny Merel theorem      369
Mazur’s theorem      97
McMillan family of integrable maps      443
McMullen’s theorem      187 367
Measure      6 36
Measure, Brolin      307
Measure, canonical on Berkovich space      129 306
Measure, Dirac      128
Measure, Haar      127
Measure, invariant      104 127 307
Measure, Lyubich      307
Measure, spherical      36
Measure, support on Galois conjugates of point      128
Measure, weak convergence      128
Merel’s theorem      97 369
Meromorphic function      244
Meromorphic function, coefficients uniquely determined      312
Meromorphic function, dynamics of      23
Meromorphic function, equi-Lipschitz family      313
Meromorphic function, equicontinuous family      254 265 313
Meromorphic function, family omitting two points      265 315
Meromorphic function, Fatou set      24 255
Meromorphic function, is continuous      253
Meromorphic function, is open      253
Meromorphic function, Julia set      24
Meromorphic function, Laurent series      245
Meromorphic function, Montel theorem      265 315
Meromorphic function, order of      244
Meromorphic function, pole      244
Meromorphic function, product is meromorphic      312
Meromorphic function, uniformly continuous family      254 313
Meromorphic function, uniformly Lipschitz      254 313
Meromorphic function, zero      244
Metric space      3
Metric space, $S^\mathbb{N}$ is a      258
Metric space, chaotic map      3
Metric space, continuous function      22
Metric space, equicontinuous collection of functions      22
Metric space, equicontinuous map      3
Metric space, Fatou set      22
Metric space, Julia set      22
Metric space, normal family of maps      25
Metric space, of sequences      258 314
Metric space, path in      306 323
Metric space, uniform continuity      39
Metric space, uniform Lipschitz      39
Metrized line bundle      410
Minimal discriminant      221
Minimal model      220 222
Minimal model, global       237
Minimal resultant      220 237
Minimal resultant, exponent at       220
Minimal resultant, is $PGL_{2}(K)$ -invariant      221
Minimal resultant, is resultant times power      222
Minimal resultant, of conjugate      219
Minimal resultant, valuation      219
Minimal Weierstrass equation      222 342
Minimal Weierstrass equation, global      343 371
Minimal Weierstrass equation, quasi      371 385
Minimal Weierstrass equation, Tate’s algorithm      342
Misiurewicz point      166
Misiurewicz point, analytic description      168 229
Misiurewicz point, dense in Mandelbrot set      166
Misiurewicz point, has bounded height      167
Misiurewicz point, in boundary of Mandelbrot set      166
Misiurewicz point, minimal polynomial      230
Misiurewicz point, no neutral cycles      166
Misiurewicz point, of type       166
Misiurewicz point, repelling critical orbit      166
Misiurewicz point, spider algorithm      168
Misiurewicz point, subhyperbolic      166
Modified Green function      318
Modular curve, dynatomic      157 158 161
Modular curve, elliptic      163 369
Modular curve, Hecke correspondence      369
Modular Jacobian      369
Moduli space      4 147
Moduli space, $\mathcal{M}^{2}\cong\mathbb{A}^{2}$       188
Moduli space, $\overline{\mathcal{M}}_{2}\cong \mathbb{P}^{2}$       194
Moduli space, coarse      160
Moduli space, degree of symmetric polynomial map      187 367 384
Moduli space, fine      160 172 230
Moduli space, isospectral family      188
Moduli space, map defined by symmetric polynomials of multipliers      187 232 367 384
Moduli space, of rational maps of degree       174
Moduli space, scheme over $\mathbb{Z}$       179 186 189
Moduli, field of      207
Monoid of Latt $\grave{e}$ s maps      382
Monomial map      29 95 325 428
Monomial map, automorphism group      234 327
Monomial map, automorphism group, in characteristic       380
Monomial map, commuting maps      326 380
Monomial map, dynamical unit      328
Monomial map, in characteristic       380
Monomial map, multiplier of periodic point      326
Monomial map, periodic point      326
Monomial map, preperiodic point      326
Monomial map, twist      206 328 380
Monomial, number of given degree      136
Montel theorem      25
Montel theorem, nonarchimedean      264 265 281
Montel theorem, of Hsia      265 271
Montel theorem, of Rivera — Letelier      311
Montel theorem, on Berkovich space      310 311
Montel theorem, with moving targets      271
Morphism      89 388
Morphism, affine      375 388
Morphism, algebraic family of      171

Morphism, bijective $\Rightarrow$ isomorphism      193
Morphism, Bogomolov conjecture      129
Morphism, canonical height associated to      99 287
Morphism, canonical measure      127
Morphism, defined over       89
Morphism, degree of      389
Morphism, degree of composition      392
Morphism, dynamical degree      397
Morphism, dynamically affine      376
Morphism, effect on absolute value      288
Morphism, effect on height      90 136 398 409
Morphism, functoriality of height for      407 421
Morphism, Green function      288
Morphism, induced on family      171
Morphism, invariant measure      127
Morphism, involution on most K3 surfaces is      417
Morphism, is open map      433
Morphism, jointly regular      397 429
Morphism, local canonical height      102
Morphism, over a variety      171
Morphism, pullback of divisor      404
Morphism, pullback of linearly equivalent divisors      405
Morphism, pushforward of divisor      404
Morton — Silverman Conjecture      96 135 368
Morton — Silverman Conjecture, for $z^{d}$       137
Morton — Silverman Conjecture, for Latt $\grave{e}$ s map      370
Moving targets      271
Multiplication map, by 2 on $\mathbb{Q}/\mathbb{Z}$       168 229
Multiplication map, on elliptic curve      31 340 383
Multiplication map, on elliptic curve, effect on height      409
Multiplication map, on elliptic curve, kernel of      343
Multiplicative group      29 30
Multiplicative group scheme      29
Multiplicative group, endomorphism      29 325
Multiplicative group, endomorphism ring      378
Multiplicative group, preperiodic points      326
Multiplicative group, quotient by $\{\pm 1\}$       29 328
Multiplicative group, torsion      326
Multiplier      18 19 47 180
Multiplier polynomial      225 227
Multiplier polynomial, resultant with cyclotomic polynomial      226
Multiplier spectrum      182 187
Multiplier spectrum, flexible Latt $\grave{e}$ s map      356
Multiplier spectrum, formal      182
Multiplier spectrum, isospectral family      188 356 362
Multiplier spectrum, of $z^{2} + bz$       182
Multiplier spectrum, of $z^{d}$       182
Multiplier spectrum, of Latt $\grave{e}$ s map      186
Multiplier, at infinity      38
Multiplier, at infinity, Latt $\grave{e}$ s map      383
Multiplier, chain rule used to calculate      19
Multiplier, Chebyshev polynomial      41
Multiplier, condition for unequal formal and primitive periods      151 226
Multiplier, conjugacy invariant      19
Multiplier, differential one-form definition      19
Multiplier, equal to one      225 230
Multiplier, flexible Latt $\grave{e}$ s map      186 358 382
Multiplier, Galois conjugates      180 184
Multiplier, integral over $\mathbb{Q}[Rat_{d}]$       180 185
Multiplier, is $PGL_{2}$ invariant      18 180 185
Multiplier, minimal polynomial      225 227
Multiplier, not equal to one      41
Multiplier, of a Latt $\grave{e}$ s map      34
Multiplier, of critical periodic point      38
Multiplier, of degree two rational map      190 233
Multiplier, of fixed point of Chebyshev polynomial      332 380
Multiplier, of fixed point of Chebyshev polynomial, in characteristic       381
Multiplier, of fixed point of Latt $\grave{e}$ s map      382
Multiplier, of fixed point of Latt $\grave{e}$ s map, of $z^{2} + bz$       182
Multiplier, rigid Latt $\grave{e}$ s map      366 384
Multiplier, root of unity      151 226
Multiplier, symmetric polynomial of      180 183 187 232 367
N $\acute{e}$ ron local height      104
Natural numbers      258
Nested sequence of closed disks      295 322
Nested sequence of closed disks, associated seminorm      297 322
Nested sequence of closed disks, equivalent      322
Neutral fixed point      322
Neutral periodic point      19
Newton polygon      248 249 275 276
Newton polygon, determines roots      249
Newton polygon, line segment      249
No wandering domains, disk conjecture      284
No wandering domains, disk theorem      284 285
No wandering domains, theorem      28 276
Nonarchimedean absolute value      44 83 242
Nonarchimedean absolute value, morphism is open map      433
Nonarchimedean analysis      242
Nonarchimedean Cauchy residue formula      314
Nonarchimedean chordal metric      see $v$

-adic chordal metric
Nonarchimedean disk component has simple form      278
Nonarchimedean hyperbolic map      279 285
Nonarchimedean inverse function theorem      312
Noncommutative monoid of Latt $\grave{e}$ s maps      382
Noncommuting involutions      410
Noncommuting involutions, three      437
Nondegenerate automorphism      226
Nonrepelling cycles, number of      27
Nonrepelling fixed point      255 314
Nonrepelling periodic point      362
Nonrepelling periodic point, in Fatou set      256
Nonsingular elliptic curve      337
Nonsingular K3 surface      438
Norm      247 288
Norm, Gauss on $\mathbb{C}_{p}[z]$       296
Norm, of a holomorphic function      247
Norm, of derivative      252
Norm, semi-      296
Norm, sup      288
Normal family      25
Normal forms lemma      190 233
Normalized coordinates      49
Normalized Markoff triple      437
Normalized rational map      51 75
Normalized valuation      44 249
Northcott’s theorem      94 407 423
Northcott’s theorem, false for rational maps      136
Notation      7
Nullstellensatz      90 92
Number field, algebraic closure      85
Number field, completion of      83
Number field, extension formula for absolute values      83 84
Number field, product formula      83
Number field, ring of -integers      83
Number field, ring of integers      83
Numerical criterion for (semi)stability      178
Octahedral group      197
One coboundary      201 202 209
One coboundary, trivial when extended      203
One cocycle      201 202
One cocycle, corresponds to twist?      203
One cocycle, determines field of moduli      209
One cocycle, extension from $Aut(\phi)$ to $PGL_{2}$       203
One cocycle, of twist      201
Open branch of Berkovich disk      300
Open disk      242
Open disk, closed under addition      243
Open disk, image of by $PGL_{2}$       317
Open disk, in $\mathbb{P}^{1}$       243
Open disk, is closed      243
Open disk, is maximal ideal      243
Open disk, rational radius      243
Open disk, standard collection      278
Open map      10 253
Open map, morphism is      433
Open map, rational map is      10 24
Open set, base of for Gel’fond topology      300
Open set, disjoint union of disk components      277
Open set, disk connected      317
Open set, iterates omit at most one point      266

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