On the entropy devil’s staircase in a family of gap-tent maps

Abstract

To analyze the trade-off between channel capacity and noise-resistance in designing dynamical systems to pursue the idea of communications with chaos, we perform a measure theoretic analysis the topological entropy function of a ‘gap-tent map’ whose invariant set is an unstable chaotic saddle of the tent map. Our model system, the ‘gap-tent map’ is a family of tent maps with a symmetric gap, which mimics the presence of noise in physical realizations of chaotic systems, and for this model, we can perform many calculations in closed form. We demonstrate that the dependence of the topological entropy on the size of the gap has a structure of the devil’s staircase. By integrating over a fractal measure, we obtain analytical, piece-wise differentiable approximations of this dependence. Applying concepts of the kneading theory we find the position and the values of the entropy for all leading entropy plateaus. Similar properties hold also for the dependence of the fractal dimension of the invariant set and the escape rate.