关于$\mathbb{R}^d$中的重叠自相似集和熵逆定理
On self-similar sets with overlaps and inverse theorems for entropy in $\mathbb{R}^d$
摘要：
研究了$\mathbb{R}^{d}$上的自相似集和自相似测度。假设定义迭代函数系统$Φ$不保持真仿射子空间，我们证明了下列之一成立：（1）维数等于平凡界（2）对于所有大的$d$，都存在来自$Φ$的图的$n$ -倍合成，它们在$n$中是超指数接近的;（3）存在$\mathbb{R}^{d}$的非平凡线性子空间，该子空间通过$Φ$的线性化而保持，并且其平移通常满足全维的集合或测度。特别地，当$Φ$的线性化不可约地作用在$\mathbb{R}^{d}$上时，维数等于$\min\{s,d\}$或者存在超指数接近的$n$ -倍组成。我们给了一些应用程序的代数系统，参数化系统，并对一些经典的例子。证明中的主要成分是$\mathbb{R}^{d}$上的测度卷积的熵增长的逆定理，以及$\mathbb{R}^{d}$上的测度与正交群上的测度卷积的熵增长。更一般地说，这一部分的文件适用于光滑行动的李群的流形。
英文摘要：
We study self-similar sets and measures on $\mathbb{R}^{d}$. Assuming that the defining iterated function system $Φ$ does not preserve a proper affine subspace, we show that one of the following holds: (1) the dimension is equal to the trivial bound (the minimum of $d$ and the similarity dimension $s$); (2) for all large $n$ there are $n$-fold compositions of maps from $Φ$ which are super-exponentially close in $n$; (3) there is a non-trivial linear subspace of $\mathbb{R}^{d}$ that is preserved by the linearization of $Φ$ and whose translates typically meet the set or measure in full dimension. In particular, when the linearization of $Φ$ acts irreducibly on $\mathbb{R}^{d}$, either the dimension is equal to $\min\{s,d\}$ or there are super-exponentially close $n$-fold compositions. We give a number of applications to algebraic systems, parametrized systems, and to some classical examples. The main ingredient in the proof is an inverse theorem for the entropy growth of convolutions of measures on $\mathbb{R}^{d}$, and the growth of entropy for the convolution of a measure on the orthogonal group with a measure on $\mathbb{R}^{d}$. More generally, this part of the paper applies to smooth actions of Lie groups on manifolds.
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