So, for example, the nice Krengel–Rokhlin formulas for ergodic maps cannot be applied directly to compute the metric entropy nor can the fundamental Adler–Konheim–McAndrew definition of topological entropy be employed. Unfortunately, the Boole maps investigated in the sequel are invariant for Lebesgue measures that are only σ-finite on noncompact Euclidean spaces. It should be noted that the finiteness of the phase space, say X, is required in the classical definitions of dynamical systems entropies: the finiteness of the measure of X-so that it can be re-scaled to a probability measure-in the case of metric entropy and finiteness of open subcoverings of X guaranteed by the compactness of the phase space for topological entropy. As we shall show, these ideas come into play in significant ways when dealing with Boole-type mappings. Moreover, ergodicity enables the derivation of precise formulas for K-S entropy for certain kinds of discrete dynamical systems, which can be used to determine the generally much harder to compute topological entropy when the two entropies can be shown to be equal. Ergodicity is another property of dynamical systems that is useful in establishing relationships between K-S and AKM entropy, such as in Theorem 2 of, and is a companion theme of this investigation. One of the more profound and rigorous connections is the variational principle stating that the topological entropy is the supremum of the metric entropy over all invariant Borel probability measures on the phase space of the dynamical system, as shown in a result that, for instance, is useful in studying weighted metric entropies (see, e.g., ). All three versions of entropy are connected in a variety of aspects, as indicated, for example, in. Topological entropy (AKM entropy) was introduced in 1965 by Adler, Konheim and McAndrew, and can be viewed as a means of providing an analog of K-S entropy for dynamical systems on topological spaces, independent of any measure-theoretic structure.
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