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

To analyze the trade-off between channel capacity and noise-resistance in d esigning dynamical systems to pursue the idea of communications with chaos, we perform a measure theoretic analysis the topological entropy function o f 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 wi th a symmetric gap, which mimics the presence of noise in physical realizat ions of chaotic systems, and for this model, we can perform many calculatio ns in closed form. We demonstrate that the dependence of the topological en tropy on the size of the gap has a structure of the devil's staircase. By i ntegrating over a fractal measure, we obtain analytical, piece-wise differe ntiable approximations of this dependence. Applying concepts of the kneadin g 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 f ractal dimension of the invariant set and the escape rate. (C) 1999 Elsevie r Science B.V. All rights reserved.