Coupled skinny baker's maps and the Kaplan-Yorke conjecture

TL;DR

This paper investigates the Kaplan-Yorke conjecture in coupled skinny baker's maps, showing that coupling can restore the equality of information and Lyapunov dimensions depending on the coupling direction.

Contribution

It demonstrates that uni-directional coupling can either restore or preserve the dimension discrepancy, depending on the coupling direction, and conjectures broader applicability.

Findings

Dimensions coincide for prevalent coupling functions in one direction.

Dimensions remain unequal for all coupling functions in the other direction.

Robust dimension discrepancy observed in certain skew-product systems.

Abstract

The Kaplan-Yorke conjecture states that for "typical" dynamical systems with a physical measure, the information dimension and the Lyapunov dimension coincide. We explore this conjecture in a neighborhood of a system for which the two dimensions do not coincide because the system consists of two uncoupled subsystems. We are interested in whether coupling "typically" restores the equality of the dimensions. The particular subsystems we consider are skinny baker's maps, and we consider uni-directional coupling. For coupling in one of the possible directions, we prove that the dimensions coincide for a prevalent set of coupling functions, but for coupling in the other direction we show that the dimensions remain unequal for all coupling functions. We conjecture that the dimensions prevalently coincide for bi-directional coupling. On the other hand, we conjecture that the phenomenon we observe for a particular class of systems with uni-directional coupling, where the information and Lyapunov dimensions differ robustly, occurs more generally for many classes of uni-directionally coupled systems (also called skew-product systems) in higher dimensions.