论文标题
典型自我诉式集的投影集和措施的尺寸
Dimensions of projected sets and measures on typical self-affine sets
论文作者
论文摘要
令$ t_1，\ ldots，t_m $为$ d \ times d $ d $可转让的真实矩阵，带有$ \ | t_i \ | <1/2 $，$ 1 \ leq i \ leq m $。对于$ {\ bf a} =（a_1，\ ldots，a_m）\ in \ bbb r^{md} $，让$π^{\ bf a}：\; σ= \ {1，\ ldots，m \}^{\ bbb n} \ to \ bbb r^d $表示与Aggine Ifs $ \ {t_ix+a_i \} _ {i = 1}}^m $相关的编码映射。我们表明，对于每一个$σ$上的$μ$ $ $ $ $ $ $ $ $ $ $均为$π^{\ bf a} _*μ$ is is Constant，对于$σ$ $σ$ $σ$的每个概率度量（\ bf a} _*μ$都是$ $ \ $ \ MATHCAL L^{md {md {md {md $ a.e. r^{md} $，其中$π^{\ bf a} _*μ$代表$π$ $ $π^{\ bf a} $的推送。特别是，我们给出了$μ$的必要条件，以便$ \ \ \ \ \ mathcal l^{md} $ - a.e.此外，对于每个分析集$ e \subsetσ$，$π^{\ bf a}（e）$的每个hausdorff，包装，下部和上部盒子计数尺寸均为$ \ mathcal l^{md l^{md} $ - a.e.给出了这些投影措施和集合的正式维度公式。估计了特殊集的Hausdorff尺寸。
Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in \Bbb R^{md}$, let $π^{\bf a}:\; Σ=\{1,\ldots, m\}^{\Bbb N}\to \Bbb R^d$ denote the coding map associated with the affine IFS $\{T_ix+a_i\}_{i=1}^m$. We show that for every Borel probability measure $μ$ on $Σ$, each of the following dimensions (lower and upper Hausdorff dimensions, lower and upper packing dimensions) of $π^{\bf a}_*μ$ is constant for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$, where $π^{\bf a}_*μ$ stands for the push-forward of $μ$ by $π^{\bf a}$. In particular, we give a necessary and sufficient condition on $μ$ so that $π^{\bf a}_*μ$ is exact dimensional for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$. Moreover, for every analytic set $E\subset Σ$, each of the Hausdorff, packing, lower and upper box-counting dimensions of $π^{\bf a}(E)$ is constant for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$. Formal dimension formulas of these projected measures and sets are given. The Hausdorff dimensions of exceptional sets are estimated.
