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Local structure of self-affine sets

Published online by Cambridge University Press:  06 September 2012

CHRISTOPH BANDT  and
ANTTI KÄENMÄKI
CHRISTOPH BANDT
Affiliation:
Institute for Mathematics and Informatics, Arndt University, 17487 Greifswald, Germany (email: [email protected])
ANTTI KÄENMÄKI
Affiliation:
Department of Mathematics and Statistics, PO Box 35 (MaD), FI-40014 University of Jyväskylä, Finland (email: [email protected])
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Abstract

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$x.$

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 5 , October 2013 , pp. 1326 - 1337
Copyright
Copyright © 2012 Cambridge University Press 

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