2020 Ian Snook Prize Problem : Three Routes to the Information Dimensions for a One-Dimensional Stochastic Random Walk and for an Equivalent Prototypical Two-Dimensional Baker Map

TL;DR

This paper explores three different methods to estimate the information dimension of a fractal model representing a nonequilibrium system, highlighting discrepancies and encouraging further analysis.

Contribution

It introduces and compares three distinct routes to determine the information dimension of a fractal model, addressing unexplained differences among estimates.

Findings

Three estimates for the information dimension are presented: 0.7897, 0.7415, 0.7337.

The paper describes three methods: Cantor-like mappings, mesh-based analysis, and Kaplan-Yorke dimension.

It emphasizes the need to understand the mechanisms behind differing estimates.

Abstract

The \$1000 Ian Snook Prize for 2020 will be awarded to the author(s) of the most interesting paper exploring a pair of relatively simple, but fractal, models of nonequilibrium systems, a dissipative time-reversible Baker Map and an equivalent stochastic random walk. The two-dimensional deterministic, time-reversible, chaotic, fractal, and dissipative Baker map is equivalent to the stochastic one-dimensional random walk model for which three distinct estimates for the information dimension, $\{ \ 0.7897,\ 0.741_5, \ 0.7337 \ \}$ have all been put forward. So far there is no cogent explanation for the differences among them. We describe the three routes to the information dimension, $D_I$: [ 1 ] iterated Cantor-like mappings, [ 2 ] mesh-based analyses of single-point iterations, and [ 3 ] the Kaplan-Yorke Lyapunov dimension, thought by many to be exact for these models. We encourage colleagues to address this Prize Problem by suggesting, testing, and analyzing mechanisms underlying these differing results.