Random Walk Equivalence to the Compressible Baker Map and the Kaplan-Yorke Approximation to Its Information Dimension

Abstract

Simple time-reversible systems can generate {\it irreversible} flows satisfying the Second Law of Thermodynamics. Maps, and equivalent random walks, can also do this. We study a pair of time-reversible Baker Maps, $N2$ and $N3$, which generate dissipative{\it fractal} phase-space structures. Steadily decreasing phase-space volumes correspond to the dissipation associated with entropy production. Like three smooth reversible dissipative one-body phase-space flows developed in the 1980s and 1990s our maps generate fractal distributions, but in two dimensions rather than three, simplifying visualization and analyses. The continuity equation, which quantifies phase-volume loss, motivates study of the fractals' reduced ``information dimensions'', which were approximated by Kaplan and Yorke in terms of two-dimensional maps' two Lyapunov exponents. The maps studied here generate fractal (fractional dimensional) distributions in their phase spaces. By mapping uniformly dense grids of points, fractal dimensions can be determined by ``area-wise'' mappings. Beginning with a uniform grid area-wise mapping of the $N2$ Baker Map provides an information dimension of 1.78969. Alternatively, as many as a trillion iterations, starting from an arbitrary point, gives a smaller ``point-wise'' dimensionality, $1.741_5$. Neither of these precisely determined estimates matches the Kaplan-Yorke conjecture value, 1.7337. In the course of studying these three different approaches to information dimension we developed random walk equivalents to both mappings, which greatly simplifies analyses. We found that for the older $ N2$ Baker map the three approaches all disagree with one another! We later discovered that for the newer $N3$ Baker mapping the three approaches to information dimension, area-wise, point-wise and Kaplan-Yorke, agree.