Dimensions of multitype Moran sets with lower limit of the contractions being zero. (English) Zbl 1437.28019
The authors study multitype Moran sets \(E\), in particular, the validity of the relation
\[
\dim_H E = s_* \leq s^* = \dim_P E = \overline{\dim}_B E,\qquad (1)
\]
where \(\dim_H\), \(\dim_P\), and \(\overline{\dim}_B\) denotes the Hausdorff, packing and upper box dimension of a set, respectively. The numbers \(s_*\) and \(s^*\) are the lower, respectively, upper pre-dimension of \(E\) (according to its natural coverings).
The above inequality (1) was established by Liu & Wen under the assumptions of primitivity and positive lower boundedness on the contracting ratios. In this article, the former assumption is relaxed to include a possible lower bound of zero of the contraction ratios. For this case, it is shown that (1) still holds (i) under a stronger assumption that primitivity, and (ii) under primitivity and two additional mild conditions.
The above inequality (1) was established by Liu & Wen under the assumptions of primitivity and positive lower boundedness on the contracting ratios. In this article, the former assumption is relaxed to include a possible lower bound of zero of the contraction ratios. For this case, it is shown that (1) still holds (i) under a stronger assumption that primitivity, and (ii) under primitivity and two additional mild conditions.
Reviewer: Peter Massopust (München)
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