Correspondence and Translation Principles for the Mandelbrot set
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چکیده
New insights into the combinatorial structure of the the Mandelbrot set are given by ‘Correspondence’ and ‘Translation’ Principles both conjectured and partially proved by E. Lau and D. Schleicher. We provide complete proofs of these principles and discuss results related to them. Stony Brook IMS Preprint #1997/14 October 1997 1. The conjectures by Lau and Schleicher Introduction. The detailed structure of the Mandelbrot set M is extremely complicated. However, much of the structure can be described by different kinds of symmetry and selfsimilarity. (For a listing of various symmetries in M , see [34].) For example, each neighborhood of a boundary point of M contains infinitely many topological copies ofM itself. This is a consequence of the (unpublished) tuning results by Douady and Hubbard (compare [25]). Whereas symmetry in the dynamic plane can mostly be explained by the action of the quadratic map, the situation is more complicated in the parameter space. Often there is a correspondence between local structure in the dynamic plane and in the parameter space which helps to understand a special symmetry in the parameter space. Typical examples are the local similarities about Misiurewicz points found by Tan Lei (see [36]): The neighborhoods of a Misiurewicz point c in the Mandelbrot set and in the corresponding Julia set are asymptotically similar in the Hausdorff metric. Roughly speaking, this provides infinitely many points in M with a ‘local rotation symmetry’. The present paper deals with symmetries whose nature is a combinatorial one. In particular, it proves two statements conjectured by Lau and Schleicher (see [23, 32]). The first of the conjectures, the Correspondence Principle, relates combinatorial structure in the dynamic
منابع مشابه
Extension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials
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تاریخ انتشار 1997