论文标题
$ l^q $ - dimension和Riemannian歧管上的自符号措施的熵维度
Existence of $L^q$-dimension and entropy dimension of self-conformal measures on Riemannian manifolds
论文作者
论文摘要
佩雷斯和Solomyak证明，在$ \ mathbb r^n $上，定义$ l^q $ - 数的极限（0，\ infty）\ setMinus \ {1 \} $，以及在不假定任何分离条件的情况下都存在自我形式度量的熵维度。通过介绍重型最大包装和分区的概念，我们证明，在加倍的度量空间上，$ l^q $ -Dimension，$ q \ in（0，\ infty）\ setMinus \ setMinus \ {1 \} $等于通用尺寸。我们还概括了$ l^q $ - 限制的存在，以与双倍属性的完整riemannian歧管上的自符号措施。特别是，这些结果适用于具有非负RICCI曲率的完整riemannian歧管。此外，假设该度量正在加倍，我们将熵维度的存在的结果扩展到完整的Riemannian歧管上的自符号度量。
Peres and Solomyak proved that on $\mathbb R^n$, the limits defining the $L^q$-dimension for any $q\in(0,\infty)\setminus\{1\}$, and the entropy dimension of a self-conformal measure exist, without assuming any separation condition. By introducing the notions of heavy maximal packings and partitions, we prove that on a doubling metric space the $L^q$-dimension, $q\in(0,\infty)\setminus\{1\}$, is equivalent to the generalized dimension. We also generalize the result on the existence of the $L^q$-dimension to self-conformal measures on complete Riemannian manifolds with the doubling property. In particular, these results hold for complete Riemannian manifolds with nonnegative Ricci curvature. Moreover, by assuming that the measure is doubling, we extend the result on the existence of the entropy dimension to self-conformal measures on complete Riemannian manifolds.
