自仿射集上平衡态的结构与仿射维数的严格单调性
Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension
摘要：
自仿射集维数论中的一个基本问题是构造高维测度，它能给出集的豪斯多夫维数的精确下界。一个自然的战略建设这样的高维措施是调查措施的最大李雅普诺夫维数，这些措施可以被解释为平衡态的奇异值功能介绍了法尔科纳。虽然这些平衡态的存在已经众所周知了一些年，但它们的结构仍然难以捉摸，特别是在大于2的维度中。本文给出了三维奇异值函数的平衡态的完整描述，特别指出所有这样的平衡态都必须是完全支持的。在高维情形下，我们也给出了这些平衡态唯一性的一个新的充分条件。作为推论，我们在三维空间中解决了一个民间传说的公开问题，证明了对于$\mathbb{R}^3$中的一个典型的自仿射集，去掉定义该集的仿射映射之一会导致豪斯多夫维数的严格降低.
英文摘要：
A fundamental problem in the dimension theory of self-affine sets is the construction of high-dimensional measures which yield sharp lower bounds for the Hausdorff dimension of the set. A natural strategy for the construction of such high-dimensional measures is to investigate measures of maximal Lyapunov dimension; these measures can be alternatively interpreted as equilibrium states of the singular value function introduced by Falconer. Whilst the existence of these equilibrium states has been well-known for some years their structure has remained elusive, particularly in dimensions higher than two. In this article we give a complete description of the equilibrium states of the singular value function in the three-dimensional case, showing in particular that all such equilibrium states must be fully supported. In higher dimensions we also give a new sufficient condition for the uniqueness of these equilibrium states. As a corollary, giving a solution to a folklore open question in dimension three, we prove that for a typical self-affine set in $\mathbb{R}^3$, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
