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Hausdorff dimension of a sub-self-affine set. (Chinese. English summary) Zbl 0960.28002
In this note, the author gives the Hausdorff dimension of a sub-self-affine set of an affine set.
Let \(S_1,\dots, S_N\) be affine compression sets on \(\mathbb{R}^n\), \[ S_i(x)= T_i(x)+ a_i,\quad i= 1,2,\dots, N, \] where \(T_i\) is an \(n\)th nonsingular matrix and \(a_i\in \mathbb{R}^n\); a compact set \(E\subset \mathbb{R}^n\) is called a sub-self-affine set of an affine set if \(E\subseteq \sum^N_{i=1} S_i(E)\), denoted by \(E(a)\), \(a= (a_1,\dots, a_N)\).
The main result is: For the Hausdorff dimension dim of \(E(a)\), we have \[ \dim(E(a))= \min\{n, d_k(T_1,\dots, T_N)\}, \] here \(d_k(T_1,\dots, T_N)\) is a special value depending on \(T_1,\dots, T_N\).
Reviewer: Su Weiyi (Nanjing)
MSC:
28A78 Hausdorff and packing measures
28A80 Fractals
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