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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