论文标题
$ l^q $ spoges上的自我措施措施
The $L^q$ spectrum of self-affine measures on sponges
论文作者
论文摘要
在本文中，$ \ mathbb {r}^d $中的海绵是一个迭代功能系统的吸引子，该功能系统由有限的严格收缩仿射图组成，其线性部分是对角线矩阵。引入了合适的分离条件，该条件在该条件下为所有$ q \ in \ Mathbb {r} $在海绵上定义的任何自动措施的$ l^q $频谱证明了变异公式。除某些特殊情况外，以前没有证明其盒子维度的存在。在某些条件下，公式具有封闭形式，通常是上限。还确定了这些措施的霜冻和盒子维度。该方法统一了几个现有结果，并将其扩展到任意维度。关键成分是引入新型压力函数，该功能旨在捕获海绵上计数数量的增长率。我们表明，这种压力满足了类似于Ledrappier的变异原理 - hausdorff尺寸的Young公式。
In this paper a sponge in $\mathbb{R}^d$ is the attractor of an iterated function system consisting of finitely many strictly contracting affine maps whose linear part is a diagonal matrix. A suitable separation condition is introduced under which a variational formula is proved for the $L^q$ spectrum of any self-affine measure defined on a sponge for all $q\in\mathbb{R}$. Apart from some special cases, even the existence of their box dimension was not proved before. Under certain conditions the formula has a closed form which in general is an upper bound. The Frostman and box dimension of these measures is also determined. The approach unifies several existing results and extends them to arbitrary dimensions. The key ingredient is the introduction of a novel pressure function which aims to capture the growth rate of box counting quantities on sponges. We show that this pressure satisfies a variational principle which resembles the Ledrappier--Young formula for Hausdorff dimension.
