The Graph Structure of Baker's Maps Implemented on a Computer

Abstract

The complex dynamics of the baker's map and its variants in infinite-precision mathematical domains and quantum settings have been extensively studied over the past five decades. However, their behavior in finite-precision digital computing remains largely unknown. This paper addresses this gap by investigating the graph structure of the generalized two-dimensional baker's map and its higher-dimensional extension, referred to as HDBM, as implemented on the discrete setting in a digital computer. We provide a rigorous analysis of how the map parameters shape the in-degree bounds and distribution within the functional graph, revealing fractal-like structures intensify as parameters approach each other and arithmetic precision increases. Furthermore, we demonstrate that recursive tree structures can characterize the functional graph structure of HDBM in a fixed-point arithmetic domain. Similar to the 2-D case, the degree of any non-leaf node in the functional graph, when implemented in the floating-point arithmetic domain, is determined solely by its last component. We also reveal the relationship between the functional graphs of HDBM across the two arithmetic domains. These findings lay the groundwork for dynamic analysis, effective control, and broader application of the baker's map and its variants in diverse domains.