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.

Understanding the underlying graph structure of a nonlinear map over a particular domain is essential in evaluating its potential for real applications. In this paper, we investigate the structure of the associated functional graph of Chebyshev permutation polynomials over a ring$\mathbb {Z}_{p^{k}}$, with p being a prime number greater than three, where every number in the ring is considered as a vertex and the existing mapping relation between two vertices is regarded as a directed edge. Based on some new properties of Chebyshev polynomials and their derivatives, we disclose how the basic structure of the functional graph evolves with respect to parameter k. First, we present a complete and explicit form of the length of a path starting from any given vertex. Then, we show that the functional graph’s strong patterns indicate that the number of cycles of any given length always remains constant as k increases. Moreover, we rigorously prove the rules on the elegant structure of the functional graph and verify them experimentally. Our results could be useful for studying the emergence mechanism of the complexity of a nonlinear map in digital computers and security analysis of its cryptographic applications.

In the past five decades, the dynamics of Baker's map in an infinite precision world have been extensively explored. However, the real structure of Baker's map when implemented in a fixed-point arithmetic domain remains unknown. This paper gives an explicit formulation for the quantized Baker's map. We then demonstrate that the maximum in-degree of the functional graph of Baker's map is invariant under any level of fixed-point arithmetic precision. Intriguingly, we observe a self-similarity phenomenon in the functional graph of a specific Baker's map with incremental increases in precision. These findings demonstrate the consistency of Baker's map across varying precision levels. This can be utilized to streamline the dynamic analysis and application design of Baker's map and its variants in finite precision environments.

Chaotic dynamics is an important source for generating pseudorandom binary sequences (PRBS). Much efforts have been devoted to obtaining period distribution of the generalized discrete Arnold's Cat map in various domains using all kinds of theoretical methods, including Hensel's lifting approach. Diagonalizing the transform matrix of the map, this article gives the explicit formulation of any iteration of the generalized Cat map. Then, its real graph (cycle) structure in any binary arithmetic domain is disclosed. The subtle rules on how the cycles (itself and its distribution) change with the arithmetic precision$e$eare elaborately investigated and proved. The regular and beautiful patterns of Cat map demonstrated in a computer adopting fixed-point arithmetics are rigorously proved and experimentally verified. The results can serve as a benchmark for studying the dynamics of the variants of the Cat map in any domain. In addition, the used methodology can be used to evaluate randomness of PRBS generated by iterating any other maps.