Keller K. — Invariant Factors, Julia Equivalences and the (Abstract)Mandelbrot Set :: Электронная библиотека попечительского совета мехмата МГУ

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Keller K. — Invariant Factors, Julia Equivalences and the (Abstract)Mandelbrot Set

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Название: Invariant Factors, Julia Equivalences and the (Abstract)Mandelbrot Set

Автор: Keller K.

Аннотация:

This book is mainly devoted to the combinatorics of quadratic holomorphic dynamics. The conceptual kernel is a self-contained abstract counterpart of connected quadratic Julia sets which is built on Thurston's concept of a quadratic invariant lamination and on symbolic descriptions of the angle-doubling map. The theory obtained is illustrated in the complex plane. It is used to give rigorous proofs of some well-known and some partially new statements on the structure of the Mandelbrot set. The text is intended for graduate students and researchers. Some elementary knowledge in topology and in functions of one complex variable is assumed.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 205

Добавлена в каталог: 17.08.2008

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Предметный указатель

$(U,{p}_{n}^{c})$ (renormalization)      162
$(X,{({f}_{i})}_{i\in I}) /\equiv\;(=\;{(X/\equiv , ({f}_{i}/\equiv))}_{i \in i})$       5
$0\langle\alpha\rangle,\;1\langle\alpha\rangle, t\langle\alpha\rangle$ (sequence tuning)      92
$Comp({\mathbb D})$ (space of compact subsets of ${\mathbb D}$ )      32
$d(\approx)$ (diameter of an equivalence relation $\approx$ on ${\mathbb T}$ )      63
$f(A)\;(=\;{\rm conv}(f(A\;\cap\;{\mathbb T})))$       32
$f/\equiv$ (factor map)      5
$Gap(\alpha)$ (infinite gap in $\mathcal B$ related to $\alpha\overline{\alpha}$ )      86
$Gap\langle\alpha \right\))$ (critical value gap of $\eth{\mathcal B}^{s}$ )      91
$Imp({\bf R}^{\beta})$ (impression of ${\bf R}^{\beta}$ )      155
$Imp({\bf R}^{\beta}_{c})$ (impression of ${\bf R}^{\beta}^{c}$ )      155
$J({p}_{c}^{n})$ (Julia set corresponding to a renormalization of ${p}_{c}$ )      14
$K({p}_{c}^{n})$ (filled-in Julia set corresponding to a renormalization of ${p}_{c}$ )      14
$PChord(({S}_{1},{S}_{2}))$ ('periodic' parameter chord associated to a dynamic pair $({S}_{1},{S}_{2})$ )      129
$X/\equiv$ (X factorized by an equivalence relation $\equiv$ )      4
$[;{k}_{1},\;{k}_{2},{k}_{3},...(,{k}_{n})]$ (continued fraction)      103
$\alpha$ structure graph      41
$\alpha$ -tuning      99
$\cong$ (substantially equal)      26
$\ddot{B}$ (preperiodic preimage of $B\;\in\;{\mathcal B}_{*}$ )      79
$\ddot{\alpha}$ (preperiodic preimage of ${\alpha}\;\in\;{{\mathbb P}_{*}}$ )      30
$\dot{\alpha}$ (periodic preimage of ${\alpha}\;\in\;{{\mathbb P}_{*}}$ )      30
$\eth{\mathcal B}^{s}$ ( $=\;\eth{\mathcal B}^{s}$ for $S\;=\;{\alpha\overline{\alpha}}$ )      86
$\eth{\mathcal L}$ (set of accumulation chords of ${\mathcal L}$ )      45
$\frac{p}{q}$ -sublimb      20
$\hat{B}$ (kneading sequence of a 'periodic' parameter chord)      79
$\hat{R}$ ('generalized' kneading sequence)      124
$\hat{\alpha}$ (kneading sequence of ${\alpha}$ )      6
$\mu({\bf a})$ (ratio between 0-length and 1-length of ${\bf a}\;\in\;{\{0,1\}}^{\mathbb N}$ )      102
$\nu({\bf a})$ (1-frequency of an ${\bf a}\;\in\;{\{0,1\}}^{\mathbb N}$ )      102
$\overline{\alpha}$ (with periodic a associated point)      53
$\overset{\curvearrowleft}{{I}^{\alpha}}(\beta)$ (counter-clockwise itinerary)      26
$\overset{\curvearrowright}{{I}^{\alpha}}(\beta)$ (clockwise itinerary)      26
$\Phi$ (uniformization map for ${\overline{\mathbb C}}\M$ )      15
$\prec$ (order on the set of Julia equivalences)      91
$\rho$ (Tent map)      185
$\sim$ (equivalence relation defining the Abstract Mandelbrot set)      16
$\widetilde{{I}^{\alpha}}(\beta)$ (specified itinerary)      62
$\widetilde{{\mathbb T}_{\bf s}^{\alpha}}$ (special subset of the circle)      30
${A}_{w*}^{\alpha}$ (special subset of the disk)      41
${B}^{(i)}$ (special chord in ${\mathcal B}_{*}\;\cup\;\{01\}$ )      81
${CARD}^{\alpha}(s)$ (number of components of ${\mathfrak G}^{\alpha}(s)$ )      41
${Comb}^{\alpha}$ (combinatorial class of M)      17
${c}^{\alpha}$ (landing point of ${\bf R}^{\alpha}$ )      149
${e}^{B}$ (characteristic symbol of a chord $B\;\in\;{\mathcal B}_{*}\;\cup\;\{01\}$ )      79
${e}^{\alpha}$ (characteristic symbol of a periodic ${\alpha}$ )      51
${Gap}_{w*}^{\alpha}$ (special infinite gap in $\eth{\mathcal B}^{s}$ )      56
${I}^{\alpha}(\beta)$ (itinerary)      6
${J}_{c}$ (Julia set)      2
${K}^{(0)}$ (small filled-in Julia set corresponding to a renormalization)      14
${K}_{c}$ (filled-in Julia set)      2
${l}_{0}^{\alpha},{l}_{1}^{\alpha},{l}_{\bf w}^{\alpha}$ (special maps on the circle)      30
${l}_{s}^{\alpha ,t},{l}_{\bf w}^{\alpha ,t}$ (special maps on the circle)      30
${Mult}_{w}$ (multiplier map on clW)      19
${m}_{0}\;\rightarrow\;{m}_{1}\;\rightarrow\;...\;\rightarrow\;{m}_{n}(\rightarrow ...)$ (internal address)      81
${p}_{c}$ (quadratic polynomial)      1
${q}^{0}$ (inversion of binary expansion)      91
${q}^{\alpha}$ (generalization of ${q}^{0}$ )      92
${q}^{\alpha}$ (map on ${\mathbb T}$ associated with ${q}^{\alpha}$ )      93
${R}_{b}$ (preperiodic chord associated with an immediately visible chord in {\mathcal B}_{*})      127
${Sturm}^{\nu}$ (set of recurrent Sturmian sequences of 1-frequency $\nu$ )      104
${Sturm}_{)}$ (special set of Sturmian sequences)      101
${SUBPER}_{n}(\gamma)$       109
${S}_{*}^{\alpha} \left( = \frac{\alpha}{2}\frac{\alpha+1}{2} \right)$       36
${S}_{s}^{\alpha}(K)$ (chord corresponding to a component K of ${\mathfrak G}^{\alpha}(s)$ )      42
${S}_{\rm w*}^{\alpha ,t}\left( = {l}_{\rm w}^{\alpha ,t}({S}_{*}^{\alpha})\right)$       38
${S}_{\rm w*}^{\alpha}\left( = {l}_{\rm w}^{\alpha}({S}_{*}^{\alpha})\right)$       36
${W}^{\alpha\overline{\alpha}}$ (hyperbolic component related to $\alpha\overline{\alpha}$ )      19
${z}_{({c}_{0},{z}_{0})}$ (periodic/preperiodic point map)      18
${z}_{c}^{\beta}$ (landing point of ${\bf R}_{c}^{\beta}$ )      10
${[A]}_{\equiv}$ (transitive hull of A with respect to $\equiv$ )      4
${[x]}_{\equiv}$ (equivalence class)      4
${\approx}^{\alpha}$ (Julia equivalence)      59
${\beta}_{1}\;\curvearrowleft\;{\beta}_{2}\;\curvearrowleft\;...\;\curvearrowleft\;{\beta}_{n}$ clockwisely ordered points in $\mathbb T$ )      33
${\beta}_{1}\;\curvearrowright\;{\beta}_{2}\;\curvearrowright\;...\;\curvearrowright\;{\beta}_{n}$ clockwisely ordered points in $\mathbb T$ )      33
${\beta}_{1}{\beta}_{2}$ (chord with end points ${\beta}_{1},{\beta}_{2}$ )      32
${\bf a}^{-}$ (0—1-word a after interchanging 0 and 1)      27
${\bf b}(\beta)$ (binary expansion of ${\beta}\;\in\;{\mathbb T}$ )      5
${\bf C}^{S}$ (curve associated to a 'parameter' chord)      149
${\bf C}^{S}_{c}$ (curve associated to a 'dynamic' chord)      146
${\bf C}^{{\alpha}_{1}{\alpha}_{2}}$ (curve defined by two parameter rays)      149
${\bf C}^{{\beta}_{1}{\beta}_{2}}_{c}$ (curve defined by two dynamic rays)      146
${\bf E}^{r}_{c}$ (equipotential curve in the dynamic plane)      157
${\bf n}_{*n}^{\gamma}$ (special periodic sequence related to renormalization)      109
${\bf R}^{\alpha}$ (parameter ray)      15
${\bf R}^{\beta}_{0}$ (standard dynamic ray)      7
${\bf R}^{\beta}_{c}$ (dynamic ray)      9
${\bf s}^{*n}$ (special symbol sequence)      107
${\bf s}_{G}$ (0—1-sequence associated with a gap G in ${\mathcal B}$ )      84
${\bf v}^{B}$ (initial subword of $\hat{B}$ of length PER(B) - 1)      79
${\bf v}^{\alpha}$ (initial subword of $\hat{\alpha}$ of length PER(a) - 1)      26
${\bf Wake}^{\alpha\overline{\alpha}}$ (wake associated to ${\alpha}\;\in\;{\mathbb P}_{*}$ )      17
${\bf w}^{n}$ (n-fold concatenation of a word w)      25
${\chi}_{s}^{t}$ (successor function)      41
${\Delta}_{utv*}^{\alpha}$ (special subset of the disk)      39
${\dot{B}}$ (periodic preimage of $B\;\in\;{\mathcal B}_{*}$ )      79
${\equiv}_{0}$ (identification associated with binary expansion)      5
${\equiv}_{1}{\circ}{\equiv}_{2}$ ('composition' of two equivalence relations)      183
${\equiv}_{2}/{\equiv}_{1}$ ('quotient' of two equivalence relations)      115 184
${\equiv}_{2}{\circ}{\equiv}_{1}$ ('composition' of two equivalence relations)      175
${\equiv}_{s}$ (special equivalence relation on ${\{0,1\}}^{\mathbb N}$ )      173
${\equiv}_{\overline{{\bf v}*}}$ (special equivalence relation on ${\{0,1\}}^{\mathbb N}$ )      173
${\hat{B}}^{+}$ (kneading sequence just behind B)      80
${\hat{B}}^{-}$ (kneading sequence just before B)      80
${\hat{\alpha}}^{\curvearrowleft}$ (counter-clockwise kneading sequence)      26
${\hat{\alpha}}^{\curvearrowright}$ (clockwise kneading sequence)      26
${\lambda}$ (empty word)      25
${\mathbb C}^{\alpha}$ ( ${\mathbb C}$ less some special parameter rays)      148
${\mathbb D}$ (unit disk)      32
${\mathbb D}^{\alpha}_{s}\;(=\;conv({\mathbb T}^{\alpha}_{s}))$       40
${\mathbb D}^{\alpha}_{\bf w}\;(=\;conv({\mathbb T}^{\alpha}_{\bf w}))$       39
${\mathbb D}^{\gamma}_{*n}\;(=\;conv({\mathbb T}^{\gamma}_{*n}))$       110
${\mathbb P}$ (points with periodic kneading sequence)      26
${\mathbb P}_{*}$ (points being periodic with respect to h)      26
${\mathbb P}_{*}^{n}$ (points in ${\mathbb P}_{*}$ , of period dividing n)      77
${\mathbb P}_{\infty}$ (non-periodic points with periodic kneading sequence)      27
${\mathbb P}_{\infty}^{\alpha}\;(=\;{\mathbb T}_{\verline{\alpha}}^{\alpha}\;for\;\alpha\;\in\;{\mathbb P}_{\infty})$       47
${\mathbb S}^{*}$ (words over the alphabet ${\mathbb S}$ )      25
${\mathbb S}^{n}$ (words in ${\mathbb S}^{*}$ of length n)      25
${\mathbb S}^{\mathbb N}$ (sequences over the alphabet ${\mathbb S}$ )      25
${\mathbb T}$ (circle)      4
${\mathbb T}\langle\alpha\rangle\;(=\;Gap\langle\alpha\rangle\;cap\;{\mathbb T})$       91
${\mathbb T}^{\alpha}_{\bf s}$ (special subset of the circle)      30
${\mathbb T}^{\alpha}_{\bf w}$ (special set of the circle)      30
${\mathbb T}^{\gamma}_{*n}$ (n-renormalization domain)      95 108
${\mathcal B}$ (closure of ${\mathcal B}_{*}$ )      84
${\mathcal B}^{S}_{*}$ ( $=\;{\mathcal B}^{\alpha}_{*}}$ for $S\;=\;\alpha\overline{\alpha}$ )      123
${\mathcal B}^{\alpha}$ (invariant lamination constructed from a 'long' chord $\frac{\alpha}{2}\frac{\alpha+1}{2}$ )      40
${\mathcal B}^{\alpha}_{*}$ (set of chords 'generating' $\eth{\mathcal B}^{\alpha};\alpha\;\in\;{{\mathbb P}_{*}}$ )      55
${\mathcal B}_{*}$ ('periodic' parameter chords)      80
${\mathcal R}^{\alpha ,t}$ (set of chords for the construction ${\mathcal B}^{\alpha}$ )      39
${\mathcal R}^{\alpha}$ (set of chords for the construction ${\mathcal B}^{\alpha}$ )      36
${\mathcal R}^{\alpha}_{i};i\;\in\;{\mathbb N}_{0}$ (sets of chords for the construction of ${\mathcal B}^{\alpha}$ )      36
${\mathfrak Bis}_{\frac{p}{q}}(W)$ (visibility tree)      22
${\mathfrak G}^{\alpha}(s)$ ( $\alpha$ structure graph)      41
${\nu}^{\alpha}(\gamma)$ ('rotation number')      105
${\overline{\bf w}}\;(=\;www\;...)$       25
${\overline{\mathbb C}}$ (Riemann sphere)      7
${\Phi}_{c}$ (Boetkher map)      7
${\phi}_{\equiv}$ (projection onto the factor)      4
${\sigma}$ (shift map)      6 25
${\sim}_{L}$ (Lavaurs' equivalence relation)      78
${\tau}_{i};i\;=\;0,1,2,..,q$ (backward-shifts)      184
${\theta}$ (conjugacy related to renormalization)      115
' (rotation by 180°)      4
0—1-sequence, recurrent      102
1-frequency      102
1-length      30
A(k) (0—1-word/0—1-sequence derived from ${\bf k}\;\in\;{\mathbb N}^{*}/{\bf k}\;\in\;{\mathbb N}^{\mathbb N})$       27
Abstract Julia set      12

Abstract Mandelbrot set      16
Address of a point      186
ALPHA fixed point      163
Angle, external      9
Angle, internal      20
Angle, n-renormalizable      108
Angle, n-renormalizable, simply      112
Backward iterate      7
Backward orbit      7
Basin, immediate      141
Behind, a chord      32
Behind, a hyperbolic component      20
Behind, the chord 01      80
BETA fixed point      163
Between, chords      32
Between, points      32
Bifurcate      20
Binary expansion      5
Boetkher map      8
Bound a gap      34
Boundary chord of a gap      34
Brjuno number      170
Bucket handle      189
Cantor set      49
Center of a hyperbolic component      169
Characteristic symbol      51
Characteristic symbol of a chord in ${\mathcal B}_{*}\;\cup\;\{01\}$       79
Chord      32
Chord, degenerate      32
Chord, narrow      137
Chord, non-degenerate      32
Chord, separating      32
Combinatorial class      17
Combinatorially smaller      78
Composant      189
Conjugacy      4
conjugate      4
Continuum      71
Contraction      185
Cremer point      142
Crossing chords      32
D (distance on ${\mathbb T}$ )      32
D(S) (length of a chord S)      32
Dendrite      71
Diameter of $\approx$       63
DPair (Q) (dynamic pair associated to a 'periodic' parameter chord Q)      130
Dynamic pair      123
Dynamic pair, parameter-like      123
Dynamic pair, semi-visible      131
Dynamic pair, visible      123
Equivalence relation, $\mathbb T,$ degenerate      12
Equivalence relation, $\mathbb T,$ non-degenerate      12
Equivalence relation, $\mathbb T,$ planar      11
Equivalence relation, closed      182
Equivalence relation, closed as a subset of the product space      4
Equivalence relation, completely f-invariant      4
Equivalence relation, completely invariant      5
Equivalence relation, f-invariant      4
Equivalence relation, invariant      5
Equivalence relation, invariant, generated      184
Factor map      5
Factor, (completely) invariant      5
Fatou component      141
Fatou component, bounded      141
Fatou set      141
Filled-in Julia set      2
Filled-in Julia set of a quadratic-like map      13
Filled-in Julia set small      14
Flow      188
Gap      34
Gap(B) (infinite gap in $\mathcal B$ related to $B\;\in\;{\mathcal B}_{*}\;\cup\;\{01\}$ )      86
Gap, critical      34
Gap, critical value      34
Gap, infinite      34
Gap, polygonal      34
Gap<S> (critical value gap in $\eth{\mathcal B}^{s}$ )      123
generation rules      184
Generation, algebraic      187
Horseshoe      189
Hybrid-equivalent      13
Hyperbolic component      18
Hyperbolic component, main      19
Impression, of a dynamic ray      155
Impression, of a parameter ray      155
Internal address      81
Internal address, angled      161
Iterate      7
Itinerary      6
Itinerary, specified      62
J(f) (Julia set of a quadratic-like map f)      13
Julia set      2
Julia set of a quadratic-like map      13
K(f) (filled-in Julia set of a quadraticlike map f)      13
Kneading sequence      6
Kneading sequence, admissible      83
Kneading sequence, just before a chord in ${\mathcal B}_{*}$       80
Kneading sequence, just behind a chord in ${\mathcal B}_{*}\;\cup\;\{01\}$       80
Kneading sequence, of a chord in ${\mathcal B}_{*}\;\cup\;\{01\}$       79
Lamination      34
Lamination, invariant      34
Land      9
Landing pattern      143
Landing point      9 15
Locally connected      10
Locally connected, at a point      10
M (Mandelbrot set)      2
Main point      65
Mandelbrot set      2
Map, angle-doubling      4
Map, hyperbolic      162
Map, polynomial-like      13
Map, quadratic-like      13
Match      189
Matchbox      189
Matchbox manifold      189
Misiurewicz point      142
Multiplier      8
Multiplier of a parameter      142
N-renormalization domain      108
Orbit      7
Orbit, periodic      7
Orbit, periodic, attractive      8
Orbit, periodic, indifferent      8
Orbit, periodic, irrationally indifferent      8
Orbit, periodic, parabolic      8
Orbit, periodic, repelling      8
Orbit, periodic, superattractive      8
Orbit, postcritical      8
Parameter, hyperbolic      18 142
Parameter, irrationally indifferent      142
Parameter, parabolic      142
Parameter, superattractive      142
Partition      27
Partition, $({k}_{1}{k}_{2}...{k}_{n})$ -regular      28
Partition, n-regular      28
Partition, n-regular derived      28
PER(B) (period of a parameter chord B      79
PER(x) (period of a point x)      7
Period, of a chord in ${\mathcal B}_{*}\;\cup\;\{01\}$       79
Period, of a hyperbolic component      18
Period, of a parameter      142
Period, of a point      7
Period, of a preperiodic point      7
Period, of an orbit      7
Point, critical      8
Point, fixed      7
Point, periodic      7
Point, periodic, associated      53
Point, periodic, attractive      8
Point, periodic, indifferent      8

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