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- W1710562010 abstract "AbstractWe define the concept of self-similarity of an object by considering endo-morphisms of the object as ‘similarity’ maps. A variety of interesting examplesof self-similar objects in geometry, algebra and arithmetic are introduced. Self-similar objects provide a framework in which, one can unite some results andconjectures in different mathematical frameworks. In some general situations,one can define a well-behaved notion of dimension for self-similar objects. Mor-phisms between self-similar objects are also defined and a categorical treatmentof this concept is provided. We conclude by some philosophical remarks. Introduction The goal of this manuscript is to study the concept of self-similarity in the abstractsense of the word. For example, an abstract notion of equality can be implementedby fixing an equivalence relation. In the same way, given any concept of similarity,we consider the associated concept of a self-similar object. For now, consider thefollowing general definition for self-similar objects:Definition 0.1 Let X be a space and let f" @default.
- W1710562010 created "2016-06-24" @default.
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- W1710562010 date "2012-11-21" @default.
- W1710562010 modified "2023-09-27" @default.
- W1710562010 title "Self-Similarity in Geometry, Algebra and Arithmetic" @default.
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