arXiv subjects
Publications and source records attributed to Chun Wei.
Let M be a b*b nonzero {0,1}-matrix. Let ρ(M) be its spectral radius and let |M^n| be the norm of its n-th iteration. In the case ρ(M)>1, we see from the spectral radius formula that {|M^n|}_{n=1}^\infty tends to \infty exponentially as n to \infty. In the case ρ(M)=1, {|M^n|}_{n=1}^\infty can be bounded or tend to \infty depending on M. The fine behavior of this sequence is completely characterized in the present paper.
Given metric spaces $E$ and $F$, it is well known that $$\dim_HE+\dim_HF\leq\dim_H(E\times F)\leq\dim_HE+\dim_PF,$$ $$\dim_HE+\dim_PF\leq \dim_P(E\times F)\leq\dim_PE+\dim_PF,$$ and $$\underline{\dim}_BE+\overline{\dim}_BF \leq\overline{\dim}_B(E\times F) \leq\overline{\dim}_BE+\overline{\dim}_BF,$$ where $\dim_HE$, $\dim_PE$, $\underline{\dim}_BE$, $\overline{\dim}_BE$ denote the Hausdorff, packing, lower box-counting, and upper box-counting dimension of $E$, respectively. In this note we shall provide examples of compact sets showing that the dimension of the product $E\times F$ may attain any of the values permitted by the above inequalities. The proof will be based on a study on dimension of the product of sets defined by digit restrictions.
We obtain a complete description for a probability measure to be doubling on an arbitrarily given uniform Cantor set. The question of which doubling measures on such a Cantor set can be extended to a doubling measure on [0; 1] is also considered.