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If $\lambda$ is the largest positive Lyapunov exponent of a piecewise linear dynamical chaotic discrete in time map, then is there a relationship between the entropy $h$ and its $\lambda$.

According to Computation of entropy and Lyapunov exponent by a shift transform by Chihiro Matsuoka and Koichi Hiraide, $h = \lambda$. And $\lambda = \log(2)$ for the tent map.

Question 1: Is the logarithm on base 2 or base 10? As entropy is in bits, it should be in base 2. However, the Lyapunov exponent (LE) for the tent map is approximately 0.69 (please correct me if wrong). If this relationship applies to all maps, then $\log_2(2)$ and $\log_{10}(2)$ would give entirely different results.

Question 2: Entropy is measured for symbols. So, when the real values of the iterations of the tent map or any other system are discretized using the concept of symbolic dynamics and represented using symbols, then will its LE change? Can I use $\log_2(2)$ or $\log_{10}(2)$ for a symbolic representation of the tent map?

Can somebody please explain what is the correct way?

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Question 1: Is the logarithm on base 2 or base 10? As entropy is in bits, it should be in base 2. However, the Lyapunov Exponent (LE) for Tent Map = 0.69 approx (please correct me if wrong). If this relationship applies to all maps, then log2(2) and log10(2) would give entirely different result.

The Lyapunov exponent is commonly defined using the exponential function $\exp$ and hence the natural logarithm $\ln$ is the correct one here. By the way: Computing the Lyapunov exponent for the tent map is very easy, as the Lyapunov exponent for maps is the average of the logarithm of the absolute value of their derivative and the derivative of the tent map is almost always $2$.

Of course, you can define the Lyapunov exponent using other bases than $e$, but that only results in a constant factor being applied to all Lyapunov exponents. Also, it’s not a big deal, since the value of a Lyapunov exponent (in contrast to its sign) becomes only interesting in relation to other Lyapunov exponents.

Question 2: Entropy is measured for symbols. So, when the real values of the iterations of the Tent Map or any other system is discretized using the concept of symbolic dynamics and represented using symbols, then will its LE change? Can I use log2(2) or log10(2) for symbolic dynamics representation of the Tent Map ?

Discretising the values of the tent map or regarding it as a bit shift is just another way of looking at the dynamics or analysing it. It does not change the actual dynamics or any of the quantities involved in the definition of the Lyapunov exponent. Thus the Lyapunov exponent does not change.

Sidenote: The paper you are referencing exclusively uses $\ln$ for the logarithm, which is unambiguously agreed upon to be the natural logarithm ($\ln$ stands for logarithmus naturalis). Had they used $\log$, there could be ambiguity, but $\ln$ is clear.

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  • $\begingroup$ Thank you, all confusions now gone. $\endgroup$ – SKM May 28 '16 at 17:51

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