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2020 Ian Snook Prize Problem : Three Routes to the Information Dimensions for One-Dimensional Stochastic Random Walks and Their Equivalent Two-Dimensional Baker Maps

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

2019

Vol. 25, No. 4

153--159

The $1000 Ian Snook Prize for 2020 will be awarded to the author(s) of the most interesting paper exploring pairs of relatively simple, but fractal, models of nonequilibrium systems, dissipative time-reversible Baker Maps and their equivalent stochastic random walks. Two-dimensional deterministic, time-reversible, chaotic, fractal, and dissipative Baker maps are equivalent to stochastic one-dimensional random walks. Three distinct estimates for the information dimension, {0.7897, 0.7415, 0.7337} have all been put forward for one such model. So far there is no cogent explanation for the differences among these estimates. We describe the three routes to the information dimension, DI : 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.

Random walks on the nonnegative integers with a left-bounded generator

Delorme C., Rinkel M.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 1

119--139

This paper studies the random walks S0 +ΣXi on the nonnegative integers, where the Xi’s are independent identically distributed random variables with generating function of type Φ(z) =Σi≥-s ciz, s a positive integer, with a convergence radius greater than 1. We infer from a link between the number of zeros of z 7→ 1 − Φ(z) inside the unit disc and inf Xi a factorisation of the symbol f(θ) = 1 − Φ(eiθ) which allows a geometrical computation of the potentials associated with these random walks. Examples illustrate this theory.

Number-conserving cellular automaton rules

Boccara N., Fukś H.

Fundamenta Informaticae

2002

Vol. 52, nr 1-3

1-13

A necessary and sufficient condition for a one-dimensional q-state n-input cellular automaton rule to be number-conserving is established. Two different forms of simpler and more visual representations of these rules are given, and their flow diagrams are determined. Various examples are presented and applications to car traffic are indicated. Two nontrivial three-state three-input self-conjugate rules have been found. They can be used to model the dynamics of random walkers.