AN ANALOGUE OF COBHAM'S THEOREM FOR FRACTALS

Description

Niveau: Secondaire, Lycée, Première
AN ANALOGUE OF COBHAM'S THEOREM FOR FRACTALS by Boris Adamczewski & Jason Bell Abstract. — We introduce the notion of k-self-similarity for compact subsets of Rn and show that it is a natural analogue of the notion of k-automatic subsets of integers. We show that various well- known fractals such as the triadic Cantor set, the Sierpinski carpet or the Menger sponge, turn out to be k-self-similar for some integers k. We then prove an analogue of Cobham's theorem for compact sets of R that are self-similar with respect to two multiplicatively independent bases k and ?; namely, we show that X is both a k- and a ?-self-similar compact subset of R if and only if it is a finite union of closed intervals with rational endpoints. 1. Introduction The notion of self-similarity is fundamental in the study of fractals. We recall (see Falconer [12]) that a compact topological space X is self-similar if there is a finite set of non-surjective homeomorphisms f1, . . . , fn : X ? X such that X = n? i=1 fi(X). It can be motivated by looking at the usual triadic Cantor set C, which is the closed subset of [0, 1] consisting of all numbers whose ternary expansion does not contain any 1s.

• self-similar compact


• self similar


• algebraic numbers only


• numbers


• state automa- ton


• automatic fractals


• words over


• all words


• ??k

Sujets

• Première
• Self-similarity
• Numbers

Informations

• Publié par: profil-ondu-2012
• Nombre de lectures: 43
• Langue: English

Extrait

ANANALOGUEOFCOBHAM’STHEOREMFORFRACTALS yb BorisAdamczewski&JasonBell Abstract .— Weintroducethenotionofk-self-similarityforcompactsubsetsofR n andshowthat itisanaturalanalogueofthenotionofk-automaticsubsetsofintegers.Weshowthatvariouswell- knownfractalssuchasthetriadicCantorset,theSierpin´skicarpetortheMengersponge,turnoutto bek-self-similarforsomeintegersk.WethenproveananalogueofCobham’stheoremforcompact setsofRthatareself-similarwithrespecttotwomultiplicativelyindependentbaseskandℓ;namely, weshowthatXisbothak-andaℓ-self-similarcompactsubsetofRifandonlyifitisafiniteunion ofclosedintervalswithrationalendpoints. 1.Introduction Thenotionofself-similarityisfundamentalinthestudyoffractals.Werecall(seeFalconer [ 12 ])thatacompacttopologicalspaceXisself-similarifthereisafinitesetofnon-surjective homeomorphismsf 1 ,...,f n :X → Xsuchthat n X=f i (X). [ 1=i ItcanbemotivatedbylookingattheusualtriadicCantorsetC,whichistheclosedsubsetof [0,1]consistingofallnumberswhoseternaryexpansiondoesnotcontainany1s.Wenotethat  C= 1 C ∪ 1 C+ 2 . 333 ThefactthatCisadisjointunionofafinitenumberofimagesofitselfunderaffinetransformations tellsusthatitisself-similar. Withthisinmind,wedefinethenotionofk-kernelforsubsetsof[0,1] d .Thek-kernelessentially looksatthepossiblesetsonecanobtainbytakingtheintersectionofXwithcertaincubesin [0,1] d withsidelength1/k a forsomepositiveintegeraandthenscalingbyafactorofk a . Definition1.1 .—GivenasubsetX⊆[0,1] d ,wedefinethek-kerneltobethecollectionof distinct  subsetsoftheform  d  (k a x 1 − b 1 ,...,k a x d − b d ) ∈ [0,1] d :(x 1 ,...,x d ) ∈ X ∩ [b j /k a ,(b j +1)/k a ], Y 1=j  wherea ≥ 0and0 ≤ b 1 ,...,b d <k a areintegers. ThefirstauthorissupportedbytheANRthroughtheproject“DyCoNum”–JCJC06134288.Healsothanks Jean-PaulAlloucheforpointingoutrelevantreferences. ThesecondauthorthanksNSERCforitsgeneroussupport.