Entropy of random chaotic interval map with noise which causes   coarse-graining
  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. 
Kouji Yano
math.PR math.DS