X. Then the variational principle for entropy states that . In general the maximum of the quantities over the set is not attained, but if additionally the entropy map is upper semicontinuous, then a measure of maximal entropy - meaning a measure in with - exists. If has a unique measure of maximal entropy , then is ergodic with respect to . Examples Let by denote the full two-sided k-shift on symbols . Let denote the partition of into cylinders of length 1. Then is a partition of for all and the number of sets is respectively. The partitions are open covers and is a topological generator. Hence . The measure-theoretic entropy of the Bernoulli -measure is also . Hence it is a measure of maximal entropy. Further on it can be shown that no other measures of maximal entropy exist. Let be an irreducible matrix with entries in and let be the corresponding subshift of finite type. Then where is the largest positive eigenvalue of . Notes See also Milnor–Thurston kneading theory For the measure of correlations in systems with topological order see Topological entanglement entropy Mean dimension References Roy Adler, Tomasz Downarowicz, Michał Misiurewicz, Topological entropy at Scholarpedia External links http://www.scholarpedia.org/article/Topological_entropy Entropy and information Ergodic theory Topological dynamics