Local structure of self-affine sets
- 6 September 2012
- journal article
- research article
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 33 (5), 1326-1337
- https://doi.org/10.1017/s0143385712000326
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 $x.$
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