The Graph Structure of Baker's maps Implemented on a Computer
Chengqing Li, Kai Tan

TL;DR
This paper investigates the finite-precision implementation of baker's maps on computers, revealing fractal patterns, structural properties, and relationships between different arithmetic domains, thus bridging the gap between theoretical dynamics and practical computation.
Contribution
It provides explicit formulations for quantized baker's maps, analyzes their structural properties across precisions, and explores their behavior in fixed-point and floating-point domains using interval arithmetic.
Findings
In-degree distribution approaches a constant with increasing precision
Fractal patterns emerge in the functional graph as precision increases
Relationship between fixed-point and floating-point baker's maps established
Abstract
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…
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Taxonomy
TopicsData Management and Algorithms · Advanced Graph Theory Research · Computational Geometry and Mesh Generation
