New results on the entropy of deterministic maps: applications to Liapunov exponent, divergence, compositional inference, and pattern encoding

By using the equation H(X, Y) = H(X) + H(Y/D), the definition of the Shannon entropy for random variables was previously extended to the entropy, without probability, or deterministic maps; then, formally, a concept of Renyi entropy for maps was obtained as a direct consequence of the former. Here, one refines the approach and generalizes these results. Shannon entropy of maps and Renyi entropy of maps are derived independently of each other but in a uniform approach, and the relations between them are exhibited. A new concept of entropy of degree c and of order s is derived, and the identification of the parameters c and s is clarified for practical applications: they refer to noisy measurement and subjectivity. Next, by using the same rationale, one derives a family of cross-entropies or divergences of maps, and then all these results are extended to deal with random maps, that is, maps that depend on a random parameter. Some illustrative applications are outlined, namely a thermodynamic approach to det...