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Inhomogeneous self-similar sets with overlaps | | Simon Baker
; Jonathan M. Fraser
; András Máthé
; | | Date: |
11 Sep 2015 | | Abstract: | It is known that if the underlying iterated function system satisfies the
open set condition, then the upper box dimension of an inhomogeneous
self-similar set is the maximum of the upper box dimensions of the homogeneous
counterpart and the condensation set. First, we prove that this ’expected
formula’ does not hold in general if there are overlaps in the construction. We
demonstrate this via two different types of counterexample: the first is a
family of overlapping inhomogeneous self-similar sets based upon Bernoulli
convolutions; and the second applies in higher dimensions and makes use of a
spectral gap property that holds for certain subgroups of $SO(d)$ for $dgeq
3$.
We also obtain new upper bounds for the upper box dimension of an
inhomogeneous self-similar set which hold in general. Moreover, our
counterexamples demonstrate that these bounds are optimal. In the final section
we show that if the emph{weak separation property} is satisfied, ie. the
overlaps are controllable, then the ’expected formula’ does hold. | | Source: | arXiv, 1509.3589 | | Services: | Forum | Review | PDF | Favorites | |
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