A Class Of K-iteration Function Systems And Their Generalization

The analysis of the fractal sets as well as the construction of the iterated functionsystems(IFSs) were firstly put forward by Hutchinson in1982. By Barnsley’s greatesteffort，the theory of IFS, has been perfecting and developing. As we know, thefoundation of fractal image compression is the basic theory of IFSs. We can generateand portray many kinds shapes of the natural world easily via IFSs，such asmountains，grasses, trees and so on. With the development of information, IFSs havebeen wildly used in the computer field, especially in the use of the compression of theimage data information. IFSs have become one of the most dynamic and wildly usefulfields, and have attracted more and more scholars’ attention. Generation of a newIFS’s attractor through constructing different kinds of IFSs to fit and approximateimages is a very significant issue in theory and application.This dissertation consists of four chapters. In the Chapter one we just recall someof the basic concepts，theorems and methods of IFSs and fractal interpolationfunctions.In the Chapter two we farther analyse the continuous dependence on theparameter of the attractor of KIFS with a parameter in Hausdorff measure on the basisof the K-IFS constructed by Sahu, and have proved the continuous dependence on theparameter of the attractor of KIFS in Hausdorff space. At last we give the character ofCollage, our result has decreased the errors compared with Sahu’s Collage Theorem. We put a numerical example which illustrates the approximation of the attractor of aKIFS to a given compact set, and the error between them is calculated.In the Chapter three we construct a class of generalized Kannan Mappings byextending the Kannan Mappings (generalized K-Mapping for short) constructed byKannan. A class of generalized KIFS consists of generalized K-Mappings is to beproved having a unique attractor. Then we analyse the continuous dependence on theparameter of the attractor of the generalized KIFS with a parameter in Hausdorffmeasure，and the result shows the continuous dependence on the parameter of theattractor of the generalized KIFS. At last we give the Collage Theorem of thegeneralized KIFS. An example about the attractor of a generalized KIFS toapproximate a given compact set is given, and the error between them is given.In Chapter four，we make a conclusion to our studies and prospect the future development of fractal studies.