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Abstract Similarity, Fractals and Chaos

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arxiv 1905.02198 v1 pith:WXKL5SZ5 submitted 2019-05-05 math.DS

Marat Akhmet , Ejaily Milad Alejaily This is my paper · ORCID
classification math.DS
keywords chaosfractalssimilarityabstractintroducedpresencecantorcases
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To prove presence of chaos for fractals, a new mathematical concept of abstract similarity is introduced. As an example, the space of symbolic strings on a finite number of symbols is proved to possess the property. Moreover, Sierpinski fractals, Koch curve as well as Cantor set satisfy the definition. A similarity map is introduced and the problem of chaos presence for the sets is solved by considering the dynamics of the map. This is true for Poincare, Li-Yorke and Devaney chaos, even in multi-dimensional cases. Original numerical simulations which illustrate the results are delivered.

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  1. Chaos on the Multi-Dimensional Cube

    math.DS 2019-08 reject novelty 3.0 of 10

    The paper claims that a shift map on sub-cube addresses makes the entire n-dimensional unit cube Devaney, Li-Yorke, and Poincare chaotic.

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