przytycki f., urbanski m. conformal fractals, dimensions and ergodic theory (book draft, cup, 2000)(252s)

przytycki f., urbanski m. conformal fractals, dimensions and ergodic theory (book draft, cup, 2000)(252s)

ID:14180421

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页数:252页

时间:2018-07-26

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1、11.11.1999Chaptersalreadyrenumbered,2b=3et CFRAFRACTAS,DESSADERGDCTHERYFeliksrzyty ki&ariuszUrbanskiThisbookisanintrodu tiontothetheoryofiterationofnon-uniformlyexpandingholomorphi mapsandtopi singeometri measuretheoryoftheunderlyinginvariantfra talsets.robabilitymeasur

2、esonthesesetsyieldinformationsonHausdor andotherfra taldimensionsandproperties.Thebookstartswitha omprehensivehapteronabstra tergodi theoryfollowedbyhaptersonuniformdistan eexpandingmapsandthermodynami alformalism.Thismaterialisappli ableinmanybran hesofdynami alsystemsandrelated elds,farb

3、eyondtheappli ationsinthisbook.opularexamplesofthefra talsetstobeinvestigatedareuliasetsforrationalfun tionsontheRiemannsphere.Thetheorywhi hwasinitiatedbyGastonulia[℄andierreFatou[F℄be ameverypopularsin ethetimewhenBenoitandelbrot'sbook[℄withbeautiful omputermadepi turesappeared.Th

4、enitbe amea eldofspe ta ulara hievementsbytopmathemati iansduringthelast20years.2Considerforexamplethemapf(z)=zfor omplexnumbersz.Thentheunit11111 ir leS=fjzj=1gisf-invariant,f(S)=S=f(S).For0;=60and2f(z)=z+,therestillexistsanf-invariantset(f) alledtheuliasetoff, lose11toS,homeomorphi t

5、oSviaahomeomorphismhsatisfyingequalityfÆh=hÆf.However(f)hasafra talshape.Forlargethe urve(f)pin hesatin nitelymanypoints;itmaypin heverywheretobe omeadendrite,oreven rumbletobe omeaCantorset.Thesesetssatisfytwomainproperties,standardattributesof" onformalfra talsets":1.Theirfra taldimens

6、ionsarestri tlylargerthanthetopologi aldimension.2.Theyare onformally"self-similar",namelyarbitrarilysmallpie eshaveshapessimilartolargepie esvia onformalmappings,hereviaiterationoff.Tomeasurefra talsetsinvariantunderholomorphi mappingsoneappliesprobabilitymeasures orrespondingtoequilibria

7、inthethermodynami alformalism.Thisisabeautifulexampleofinterla ingofideasfrommathemati sandphysi s.Aprototypelemma[B,emma1.1℄attherootsofthethermodynami alformalismsaysthatforgivenrealnumbersa;:::;athequantity1nnnXXF(p;:::p)=plogp+p1niiiii=1i=1nihasmaximu

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