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

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 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.