Combinatorics of distance doubling maps

Karsten Keller; Steifen Winter

doi:10.4064/fm187-1-1

Combinatorics of distance doubling maps

Karsten Keller^*, Steifen Winter

^*Corresponding author for this work

Institute of Mathematics

Abstract

We study the combinatorics of distance doubling maps on the circle ℝ/ℤ with prototypes h(β) = 2β mod 1 and h̄(β) = -2β mod 1, representing the orientation preserving and orientation reversing case, respectively. In particular, we identify parts of the circle where the iterates f^on of a distance doubling map f exhibit "distance doubling behavior". The results include well known statements for h related to the structure of the Mandelbrot set M. For h̄ they suggest some analogies to the structure of the tricorn, the " antiholomorphic Mandelbrot set".

Original language	English
Journal	Fundamenta Mathematicae
Volume	187
Issue number	1
Pages (from-to)	1-35
Number of pages	35
ISSN	0016-2736
DOIs	https://doi.org/10.4064/fm187-1-1
Publication status	Published - 01.01.2005

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AB - We study the combinatorics of distance doubling maps on the circle ℝ/ℤ with prototypes h(β) = 2β mod 1 and h̄(β) = -2β mod 1, representing the orientation preserving and orientation reversing case, respectively. In particular, we identify parts of the circle where the iterates fon of a distance doubling map f exhibit "distance doubling behavior". The results include well known statements for h related to the structure of the Mandelbrot set M. For h̄ they suggest some analogies to the structure of the tricorn, the " antiholomorphic Mandelbrot set".

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Combinatorics of distance doubling maps

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