Local structure of self-affine sets

The structure of a self-similar set with the open set condition does not change under magnification. For self-affine sets, the situation is completely different. We consider self-affine Cantor sets $E\subset \mathbb {R}^2$ of the type studied by Bedford, McMullen, Gatzouras and Lalley, for which the...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2013-10, Vol.33 (5), p.1326-1337
Main Authors: BANDT, CHRISTOPH, KÄENMÄKI, ANTTI
Format: Article
Language:English
Subjects:
Approximation
Dynamical systems
Fibers
Intervals
Projection
Scenery
Self-similarity
Tangents
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Summary:The structure of a self-similar set with the open set condition does not change under magnification. For self-affine sets, the situation is completely different. We consider self-affine Cantor sets $E\subset \mathbb {R}^2$ of the type studied by Bedford, McMullen, Gatzouras and Lalley, for which the projection onto the horizontal axis is an interval. We show that in small square $\varepsilon $-neighborhoods $N$ of almost each point $x$ in $E,$ with respect to many Bernoulli measures on the address space, $E\cap N$ is well approximated by product sets $[0,1]\times C$, where $C$ is a Cantor set. Even though $E$ is totally disconnected, all tangent sets have a product structure with interval fibers, reminiscent of the view of attractors of chaotic differentiable dynamical systems. We also prove that $E$has uniformly scaling scenery in the sense of Furstenberg, Gavish and Hochman: the family of tangent sets is the same at almost all points$x.$
ISSN:0143-3857
1469-4417
DOI:10.1017/S0143385712000326