# Kolmogorov entropy and noise

There are various ways of calculating the Kolmogorov entropy (KE)of dynamical system. According to Pesin's theorum, it is the summation of the lyapunov exponents; frominformation theoretic concet, it is the supremum of the Shannon entropy defines for all the partitions. I am using the formula given here.

(Jaap C. Schouten, Floris Takens, and Cor M. van den Bleek , " Maximum-likelihood estimation of the entropy of an attractor" Phys. Rev. E 49, 126)

The desired signal $x(t)$ whose entropy is to be measured is corrupted with additive white gaussian noise. Say: $$x(t)= ax(t-1) + bx(t-1) + \text{Chaos_Signal}+\text{White noise}$$ I want to calculate the KE of $x$. Assuming, the chaotic signal to be logistic map, my question are

(Q1) for different values of the parameters $(a_1,b_1),(a_2,b_2),\ldots$ will the KE measure be same for a particular signal-to-noise ratio (SNR)?

(Q2) Will KE decrease with increase of SNR? Intuitively, entropy is the measure of randomness. So, higher the level of noise (low SNR) then greater is the entropy. Again, noise also changes the value of the Lyapunov exponents of the system. I guess, Lyapunov exponent increases. Is there any proof that shows relationship between KE of time series increasing or decreasing with noise?

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