In this master thesis we investigate, from a topological point of view and without applying Thurston´s Theorem, why the mating of the so called basilica polynomial <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-1%7D(z)=z%5E%7B2%7D-1" /> and the dendrite <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7Bi%7D(z)=z%5E%7B2%7D+i" /> is shared with the mating of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-1%7D" /> and the dendrite <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-i%7D(z)=z%5E%7B2%7D-i" />. Both these matings equal the rational map <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?R_%7B3%7D(z)=%5Cfrac%7B3%7D%7Bz%5E%7B2%7D+2z%7D" />. Defined in the thesis are for both matings homeomorphic changes of coordinates<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cpsi_%7B-1%7D%5E%7B%5Cpm%7D" /> from the set <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?L=%5Coverset%7B%5Ccirc%7D%7BK%7D%5Cleft(f_%7B-1%7D%20%5Cright)%5Ccup%5Cleft(%5Ccup_%7Bn=0%7D%5E%7B%5Cinfty%7Df_%7B-1%7D%5E%7B%5Ccirc(-n)%7D(z_%7B%5Calpha%7D)%5Cright)" /> to the Fatou and Julia set of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?R_%7B3%7D" />. Here <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?K%5Cleft(f_%7B-1%7D%20%5Cright)" /> is the filled Julia set of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-1%7D" /> and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?z_%7B%5Calpha%7D" /> is the <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" />-fixed point of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?K%5Cleft(f_%7B-1%7D%20%5Cright)" />. / I detta examensarbete undersöker vi, från en topologisk synvinkel och utan applicering av Thurstons teorem, varför matchningen av det så kallade basilikapolynomet <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-1%7D(z)=z%5E%7B2%7D-1" /> och dendriten <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7Bi%7D(z)=z%5E%7B2%7D+i" /> är delad med matchningen av <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-1%7D" /> och dendriten <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-i%7D(z)=z%5E%7B2%7D-i" />. Båda dessa matchningar är lika med den rationella avbildningen <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?R_%7B3%7D(z)=%5Cfrac%7B3%7D%7Bz%5E%7B2%7D+2z%7D" />. Definierat i examensarbetet är för båda matchningarna homoemorfa koordinatbyten<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cpsi_%7B-1%7D%5E%7B%5Cpm%7D" /> från mängden<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?L=%5Coverset%7B%5Ccirc%7D%7BK%7D%5Cleft(f_%7B-1%7D%20%5Cright)%5Ccup%5Cleft(%5Ccup_%7Bn=0%7D%5E%7B%5Cinfty%7Df_%7B-1%7D%5E%7B%5Ccirc(-n)%7D(z_%7B%5Calpha%7D)%5Cright)" /> till Fatou- och Juliamängden av <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?R_%7B3%7D" />. Här är <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?K%5Cleft(f_%7B-1%7D%20%5Cright)" /> den ifyllda Juliamängden av avbildningen <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f_%7B-1%7D" /> och <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?z_%7B%5Calpha%7D" /> är den <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" />-fixerade punkten i <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?K%5Cleft(f_%7B-1%7D%20%5Cright)" />.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:umu-96855 |
| Date | January 2014 |
| Creators | Bjørnstad Pedersen, Lars |
| Publisher | Umeå universitet, Institutionen för fysik |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Student thesis, info:eu-repo/semantics/bachelorThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
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