Hausdorff Dimension of Boundaries of Self-affine Tiles in Rn
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Hausdorff Dimension of Boundaries of Self-affine Tiles in R
We present a new method to calculate the Hausdorff dimension of a certain class of fractals: boundaries of self-affine tiles. Among the interesting aspects are that even if the affine contraction underlying the iterated function system is not conjugated to a similarity we obtain an upperand and lower-bound for its Hausdorff dimension. In fact, we obtain the exact value for the dimension if the ...
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Overlap coincidence in a self-affine tiling in R is equivalent to pure point dynamical spectrum of the tiling dynamical system. We interpret the overlap coincidence in the setting of substitution Delone set in R and find an efficient algorithm to check the pure point dynamical spectrum. This algorithm is easy to implement into a computer program. We give the program and apply it to several exam...
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Simple sufficient conditions are given for the validity of a formula of Falconer [3] describing the Hausdorff dimension of a self-affine set. These conditions are natural (and easily checked) geometric restrictions on the actions of the affine mappings determining the self-affine set. It is also shown that under these hypotheses the self-affine set supports an invariant Gibbs measure whose Haus...
متن کاملSelf-Affine Tiles in Rn
A self-affine tile in R is a set T of positive measure with A(T) = d ∈ $ < (T + d), where A is an expanding n × n real matrix with det (A) = m on integer, and $ = {d 1 ,d 2 , . . . , d m } ⊆ R is a set of m digits. It is known that self-affine tiles always give tilings of R by translation. This paper extends the known characterization of digit sets $ yielding self-affine tiles. It proves seve...
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We prove that generically, for a self-affine set in R, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension. This gives a partial positive answer to a folklore open question.
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تاریخ انتشار 1998