A new information theoretic approach to the entropy of non-random discrete maps relation to fractal dimension and temperature of curves

By combining the informational meaning of the quantity ln¦k¦, where k is a constant, together with the maximum conditional entropy principle, one can obtain a new family of entropies (pattern entropies), depending upon a real valued parameter, for non-random functions. This entropy is fully consistent with the entropy of random variables on the one hand and fractal dimension on the other; and, moreover, it contains the Liapunov exponent as a special case. It is really an entropy of form and, more exactly, appears to be the entropy of a function given a scanning frequency distribution. Some examples are given which exhibit its genuine physical meaning. By using a formal identification, suggested by Gibbsʹ distribution, one arrives at a modelling for the temperature of maps.