计算机上实现的Baker映射的图结构
The Graph Structure of Baker's maps Implemented on a Computer
AI总结：
针对有限精度计算机上Baker映射真实结构不明确的问题，本文给出量化Baker映射及其高维扩展的显式公式，揭示其功能图的入度特性、分形与半分形模式，建立了定点与浮点域功能图的关联，为相关动力学分析与应用奠定基础。
摘要：
过去五十年间，学界已对无限精度数学领域中Baker映射及其变体的复杂动力学特性开展了大量分析。然而，它们在有限精度计算机上实现的真实结构仍不明确。本文给出了量化Baker映射及其高维扩展的显式公式。研究揭示了若干特性：例如随着精度提升，状态映射网络的入度分布趋近于一个常数，且在不同精度的定点算术下，最大入度保持一致。我们还观察到，随着精度提升，Baker映射的功能图呈现出以分形维数为特征的分形模式。随后，我们深入研究了高维Baker映射（HDBM）在定点和浮点算术域中生成的功能图的结构细节。本研究的一个有趣之处是使用区间算术建立了HDBM在这两个计算域中功能图之间的关联。一项尤为引人关注的发现是，随着精度逐步提升，某一特定Baker映射变体的功能图中出现了半分形模式。本研究所得见解为Baker映射及其变体在各领域的动力学分析与应用提供了基础认知。
英文摘要：
The complex dynamics of baker's map and its variants in an infinite-precision mathematical domain have been extensively analyzed in the past five decades. However, their real structure implemented in a finite-precision computer remains unclear. This paper gives an explicit formulation for the quantized baker's map and its extension into higher dimensions. Our study reveals certain properties, such as the in-degree distribution in the state-mapping network approaching a constant with increasing precision, and a consistent maximum in-degree across various levels of fixed-point arithmetic precision. We also observe a fractal pattern in baker's map functional graph as precision increases, characterized by fractal dimensions. We then thoroughly examine the structural nuances of functional graphs created by the higher-dimensional baker's map (HDBM) in both fixed-point and floating-point arithmetic domains. An interesting aspect of our study is the use of interval arithmetic to establish a relationship between the HDBM's functional graphs across these two computational domains. A particularly intriguing discovery is the emergence of a semi-fractal pattern within the functional graph of a specific baker's map variant, observed as the precision is incrementally increased. The insights gained from our research offer a foundational understanding for the dynamic analysis and application of baker's map and its variants in various domains.
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