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Coboundary on colored tiling space as Rauzy fractal. (English) Zbl 1024.37008

Summary: We consider the space \(\Omega\) of colored tilings corresponding to a weighted substitution introduced by the second author [Isr. J. Math. 106, 313-337 (1998; Zbl 0914.28014)], which is a kind of natural extension of the \(f\)-expansion for a piecewise linear \(f\). We give a characterization of adapted coboundaries, which are \(\alpha\)-\(G\)-homogeneous on the space of integer points \(\Omega_0\), where \(\alpha\) is a complex number with a negative real part. The image of such a coboundary corresponding to a weighted substitution of cubic Pisot type, is a fractal set called Rauzy fractal. We also consider the Fibonacci tiling and the \(\alpha\)-\(G\)-homogeneous, adapted coboundary on it. This coboundary together with the fractional part give a geometrical representation of the Fibonacci expansions that is more or less known.

MSC:

37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)

Citations:

Zbl 0914.28014
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Full Text: DOI

References:

[1] Kamae, Teturo, Linear expansions, strictly ergodic homogeneous cocycles and fractals, Israel Journal of Mathematics, 106, 313-337 (1998) · Zbl 0914.28014
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