Entropy of random chaotic interval map with noise which causes coarse-graining

TL;DR

This paper investigates how noise-induced coarse-graining affects the entropy of chaotic interval maps, showing that the Shannon entropy of the resulting Markov chain converges to a finite limit often exceeding the original map's Lyapunov exponent.

Contribution

It proves the existence of a finite limit for the Shannon entropy of coarse-grained Markov chains derived from chaotic maps with noise, revealing a relationship between noise, entropy, and Lyapunov exponents.

Findings

Shannon entropy converges to a finite limit as noise vanishes.

The limit often exceeds the original map's Lyapunov exponent.

The results apply to maps topologically conjugate to piecewise-linear maps with ergodic Lebesgue measure.

Abstract

A random chaotic interval map with noise which causes coarse-graining induces a finite-state Markov chain. For a map topologically conjugate to a piecewise-linear map with the Lebesgue measure being ergodic, we prove that the Shannon entropy for the induced Markov chain possesses a finite limit as the noise level tends to zero. In most cases, the limit turns out to be strictly greater than the Lyapunov exponent of the original map without noise.