Topological properties of Rauzy fractals:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Paris
Soc. Math. de France
2009 [erschienen] 2010
|
| Schriftenreihe: | Mémoire de la Société Mathématique de France
118 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | 140 S. graph. Darst. |
| ISBN: | 9782856292907 |
Internformat
MARC
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| 100 | 1 | |a Siegel, Anne |d 1975- |e Verfasser |0 (DE-588)130150940 |4 aut | |
| 245 | 1 | 0 | |a Topological properties of Rauzy fractals |c Anne Siegel ; Jörg M. Thuswaldner |
| 264 | 1 | |a Paris |b Soc. Math. de France |c 2009 [erschienen] 2010 | |
| 300 | |a 140 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Datensatz im Suchindex
| _version_ | 1817704621903183872 |
|---|---|
| adam_text |
CONTENTS
1.
Introduction
. 7
1.1.
The role of substitutions in several branches of mathematics
. 7
1.1.1.
Combinatorics
. 7
1.1.2.
Number theory
. 8
1.1.3.
Dynamical systems
. 9
1.1.4.
Applications to tiling theory, theoretical physics and discrete
geometry
. 10
1.2.
The geometry of substitutions: Rauzy fractals
. 12
1.2.1.
The Rauzy fractal for the Tribonacci substitution
. 12
1.2.2.
Using a Rauzy fractal and its topological properties
. 13
1.3.
Topological properties of central tiles
. 14
2.
Substitutions, central tiles and beta-numeration
. 19
2.1.
Substitutions
. 19
2.1.1.
General setting
. 19
2.1.2.
Pisot substitutions
. 20
2.2.
The central tile associated with a unit Pisot substitution
. 20
2.2.1.
A broken line associated with a Pisot substitution
. 21
2.2.2.
A suitable decomposition of the space
. 21
2.2.3.
Definition of the central tile
. 22
2.3.
Central tiles viewed as graph directed iterated function systems
. 23
2.3.1.
Disjointness of the
subtiles
of the central tile
. 25
2.4.
Examples of central tiles and their
subtiles . 25
2.5.
Recovering beta-numeration from unit Pisot substitutions
. 28
2.6.
Dumont-Thomas numeration
. 32
3.
Multiple tilings induced by the central tile and its
subtiles . 35
3.1.
The self-replicating multiple tiling
. 36
3.1.1.
The self-replicating translation set
. 36
3.1.2.
The dual substitution E^ and the geometric property (F)
. 36
3.1.3.
Definition of the self-replicating multiple tiling
. 38
3.2.
The lattice multiple tiling
. 39
3.3.
The tiling property and the Pisot conjecture
. 43
CONTENTS
4. Statement
of the main results: topological properties of central tiles
45
4.1.
A description of specific subsets of the central tile
. 45
4.2.
Tiling properties of the central tile and its
subtiles . 47
4.3.
Dimension of the boundary of central tiles
. 48
4.4.
Exclusive inner points and the geometric property (F)
. 49
4.5.
Connectivity properties of the central tile
. 50
4.6.
Disklikeness of the central tile and its
subtiles . 51
4.7.
The fundamental group of the central tile and its
subtiles . 52
5.
Graphs that contain topological information on the central tile
. 53
5.1.
The graph detecting expansions of zero
. 53
5.2.
Graphs describing intersections of two
subtiles . 55
5.2.1.
The self-replicating boundary graph
. 58
5.2.2.
The lattice boundary graph
. 61
5.3.
Graphs related to the connectivity of the central tile
. 63
5.4.
Contact graphs
. 65
5.5.
Triple points and connectivity of the boundary
. 70
5.6.
Quadruple points and connectivity of pieces of the boundary
. 73
5.7.
Application of the graphs to the disklikeness criterion
. 76
6.
Exact statements
and proofs of the main results
. 81
6.1.
Tiling properties
. 81
6.2.
Dimension of the boundary of the
subtiles
C7(i)
(¡ЕЙ)
. 84
6.3.
Inner points of
7
and the geometric property (F)
. 85
6.4.
Connectivity
. 86
6.5.
Homeomorphy to a closed disk
. 87
6.5.1.
A necessary condition coming from the lattice tiling property
. 87
6.5.2.
Preliminary results on GIFS
. 88
6.5.3.
A sufficient condition for the
subtiles
cJ{i)
(і Є Й)
to be
disklike
. 91
6.6.
The fundamental group
. 95
7.
Technical proofs and definitions
.109
7.1.
A technical proof from Chapter
3 .109
7.2.
Technical proofs from Chapter
5 .
Ill
7.3.
Details for the quadruple point graph
.117
8.
Perspectives
.121
8.1.
Topology
.121
8.2.
Number theory
.122
8.3.
Invariants in dynamics and geometry
.124
8.4.
Effective constructions and generalizations
.127
Bibliography
.129
MEMOIRES DE LA
SMF
118
Rauzy
fractals play a major role in many branches of mathematics
including number theory, dynamical systems, combinatorics and the
theory of quasicrystals. In many problems the topological structure
of Rauzy fractals is of crucial importance. This monograph contains
a systematic study of topological properties of Rauzy fractals. After a
thorough survey on basic results on these fractals we aim at collecting
known results on the topology of Rauzy fractals an prove a variety of
new ones. We investigate tiling properties, connectivity, homeomorphy
to a closed disk as well as the fundamental group of of Rauzy fractals.
Our main tools are results from plane topology and certain classes of
graphs that are defined in terms of tiling properties of Rauzy fractals.
We illustrate all our results with many examples.
The monograph ends with a chapter containing perspectives on
further research related to this topic.
Les
fractals
de
Rauzy
apparaissent dans diverses
branches
des mathéma¬
tiques telles que la théorie des nombres, les systèmes dynamiques, la com-
binatoire et la théorie des quasi-cristaux. De nombreuses questions font
alors intervenir la structure topologique des
fractals.
Cette monographie
propose une étude systématique des propriétés topologiques des
fractals
de Rauzy. Les premiers chapitres de ce document rappellent les enjeux
mathématiques relatifs aux
fractals
de Rauzy ainsi que les principaux
résultats connus à leur sujet. Sont ensuite discutés des propriétés de pa¬
vages, de connexité, d'homéomorphisme à un disque, ainsi que le groupe
fondamental de ces ensembles. Les méthodes s'appuient sur des résul¬
tats en
topologie
du plan et sur la construction de graphes pour décrire
la structure des pavages associés aux
fractals.
De nombreux exemples
caractéristiques sont présentés. Un chapitre final discute des principales
perspectives de recherches liées à cette thématique. |
| any_adam_object | 1 |
| author | Siegel, Anne 1975- Thuswaldner, Jörg M. |
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| id | DE-604.BV036854929 |
| illustrated | Illustrated |
| indexdate | 2024-12-06T15:16:52Z |
| institution | BVB |
| isbn | 9782856292907 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020770821 |
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| physical | 140 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Soc. Math. de France |
| record_format | marc |
| series2 | Mémoire de la Société Mathématique de France |
| spelling | Siegel, Anne 1975- Verfasser (DE-588)130150940 aut Topological properties of Rauzy fractals Anne Siegel ; Jörg M. Thuswaldner Paris Soc. Math. de France 2009 [erschienen] 2010 140 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mémoire de la Société Mathématique de France 118 Fraktal (DE-588)4123220-3 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Fraktal (DE-588)4123220-3 s Topologie (DE-588)4060425-1 s DE-604 Thuswaldner, Jörg M. Verfasser (DE-588)142994758 aut Société Mathématique de France Mémoire de la Société Mathématique de France 118 (DE-604)BV000000921 118 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020770821&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020770821&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Siegel, Anne 1975- Thuswaldner, Jörg M. Topological properties of Rauzy fractals Fraktal (DE-588)4123220-3 gnd Topologie (DE-588)4060425-1 gnd |
| subject_GND | (DE-588)4123220-3 (DE-588)4060425-1 |
| title | Topological properties of Rauzy fractals |
| title_auth | Topological properties of Rauzy fractals |
| title_exact_search | Topological properties of Rauzy fractals |
| title_full | Topological properties of Rauzy fractals Anne Siegel ; Jörg M. Thuswaldner |
| title_fullStr | Topological properties of Rauzy fractals Anne Siegel ; Jörg M. Thuswaldner |
| title_full_unstemmed | Topological properties of Rauzy fractals Anne Siegel ; Jörg M. Thuswaldner |
| title_short | Topological properties of Rauzy fractals |
| title_sort | topological properties of rauzy fractals |
| topic | Fraktal (DE-588)4123220-3 gnd Topologie (DE-588)4060425-1 gnd |
| topic_facet | Fraktal Topologie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020770821&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020770821&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000921 |
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