发表机构
Gdańsk University of Technology; Tarbiat Modares University（格但斯克理工大学; 塔比阿特莫达雷斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对区间平移映射，结合熵证明其几乎处处可逆，利用Arnoux-Ornstein-Weiss定理构造可数IET模型，通过缺陷集刻画有限/可数IET模型的几何特征，并显式给出Bruin-Troubetzkoy映射的可数IET模型。
AI 中文摘要
区间平移映射是一种分段平移映射，其不同连续区间的像可能发生重叠。本文研究其测度动力学何时可由有限或可数区间交换变换（IET）表示。首先，我们基于熵给出了一个直接证明，该证明对应一个已知结论：对于每个非原子不变概率测度，区间平移映射在几乎处处可逆，证明过程利用了自然分支编码复杂度的多项式上界。接着，我们应用Arnoux-Ornstein-Weiss定理的一个推论：标准概率空间上的每个非原子保测自同构都存在一个可数区间交换变换模型，将其应用于几乎处处可逆的核心，可得到每个测度区间平移映射的抽象可数区间交换变换模型。随后，我们研究由不变测度的分布函数定义的保序规范坐标，对每个分支引入正缺陷测度，记录其直接像携带的过剩测度；该缺陷测度向分布坐标的推送给出了一个规范割集，在该割集之外，诱导映射局部为平移。有限缺陷支撑对应有限区间交换变换模型，而勒贝格零测度的缺陷割集对应可数模型。在分支非奇异条件下，缺陷支撑是支撑中活跃间隙产生的割集的闭包，为有限情形提供了等价几何刻画。最后，针对自相似无穷型Bruin-Troubetzkoy映射，我们显式计算了规范缺陷割集，其为可数无穷且勒贝格零测度，对应一个真正可数、非有限的区间交换变换模型。
英文摘要
Interval translation maps are piecewise translations for which the images of distinct continuity intervals may overlap. We study when their measured dynamics can be represented by finite or countable interval exchange transformations. First, we give a direct entropy-based proof of the known fact that an interval translation map is invertible almost everywhere with respect to every nonatomic invariant probability measure. The proof uses a polynomial upper bound for the complexity of the natural branch coding. We then use a consequence of a theorem of Arnoux, Ornstein, and Weiss: every nonatomic measure-preserving automorphism of a standard probability space admits a countable interval exchange transformation model. Applied to the almost-everywhere invertible core, this gives an abstract countable interval exchange transformation model for every measured interval translation map. We next study the canonical order-preserving coordinate defined by the distribution function of the invariant measure. For each branch, we introduce a positive defect measure recording the excess measure carried by its direct image. Its push-forward to the distribution coordinate gives a canonical cut set such that, away from this set, the induced map is locally a translation. Finite defect support yields a finite interval exchange transformation model, while a Lebesgue-null defect cut set yields a countable model. Under branchwise nonsingularity, the defect support is the closure of the cuts arising from active gaps in the support, giving an equivalent geometric characterization of the finite case. Finally, for a self-similar infinite-type Bruin-Troubetzkoy map, we compute the canonical defect cut set explicitly. It is countably infinite and Lebesgue-null, yielding a genuinely countable, non-finite interval exchange transformation model.
Comments63 pages
