Fractals And Universal Spaces In Dimension Theory

Historically for metric spaces the quest for universal spaces in dimensiontheory spanned approximately a century of mathematical research. The historybreaks nat... » 

Historically for metric spaces the quest for universal spaces in dimensiontheory spanned approximately a century of mathematical research. The historybreaks naturally into two periods the classical separable metric and themodern notnecessarily separable metric. The classical theory is now welldocumented in several books. This monograph is the first book to unify themodern theory from 19602007. Like the classical theory the modern theoryfundamentally involves the unit interval. Unique features include The use ofgraphics to illustrate the fractal view of these spaces Lucid coverage of arange of topics including pointset topology and mapping theory fractalgeometry and algebraic topology A final chapter contains surveys andprovides historical context for related research that includes other imbeddingtheorems graph theory and closed imbeddings Each chapter contains a commentsection that provides historical context with references that serve as a bridgeto the literature. This monograph will be useful to topologists tomathematicians working in fractal geometry and to historians of mathematics.Being the first monograph to focus on the connection between generalizedfractals and universal spaces in dimension theory it will be a natural textfor graduate seminars or selfstudy the interested reader will find manyrelevant open problems which will create further research into these topics.TOCPreface. Introduction. Construction of JAj alpha. SelfSimilarity andJn1 for Finite n. NoCarry Property of wA. Imbedding Ja in Hilbert Space.Infinite IFS with Attractor wA. Infinite IFS with Attractor wA. DimensionZero. Decompositions. The Jn1 Imbedding Theorem. MinimalExponentQuestion. The JA Imbedding Theorem. 19922007 Ja Related Research. IsotopyMoves J5 Into 3Space. From 2Web IFS to 2 Simplex IFS 2Space and the 1Sphere. From 3Web IFS to 3Simplex 3Space and the 2Sphere. BackgroundBasics. The Standard Simplex. Measures and Fractal Dimension. Bibliography.Index. «