What is the connectivity between Boltzmann's entropy expression and Shannon's entropy expression? mentions a realtionship between Shannon entropy and Bolltzmann entropy. Is there a relationship between Kolmogorov Sinai entropy and Boltzmann entropy? And kolmogorov entropy and Shannon entropy?
This is a contentious question depending on the researcher answering and their background. Much of the available literature on the subject(s) will say that although the forms of these equations are similar, there is no relation between Shannon and Boltzman entropy. The best treatment I've seen stems from E.T. Jaynes' classical paper on the subject and is presented by Arieh Ben-Naim in his book Farewell to Entropy.