1 1 four boxes of M; are occupied, the Mz; = rf indicates that all four kinds of
transitions are possible. Since we remain in the context of a 0,1, two state system, the growth rate of the possible equals the logarithm, base two, of the sum of the entries in the boxes of the left-top-row to right-bottom-row diagonal called the trace and Hr = log (1 + 1) = log2(2) = 1. Consistent with intuition, since every transition is possible, the topological entropy of M; as indicated in its M;; is maximal (= 1). Another expression equal to the sum of the trace (the sum of the upper left to lower right diagonal) in a square matrix, is its leading eigenva/ue, most often symbolized with a lambda, 41. The logarithm of the leading eigenvalue of the transition incidence matrix is equal to its topological entropy. Symbolically, Hz (Mz) = logz (A1 ) = log2 (2) = 1. Standard elementary linear algebra texts describe how to compute eigenvalues, these relations and related operations as well as their foundational theorems.
Before computing the entropy of the distribution of probabilities among the
possibles as the metric entropy, Hy, let us notice again that the occupancies in the
1 3 four entry boxes of the transition matrix M; are not uniform, M; = 3 9° This leads
naturally to the intuition that for this series of binary transitions, Hy, in contrast with Hy, will not be maximal, i.e. not equal to 1 and the nonuniformity of H; and Hy is a computational expression of what we mean by a state of in-between entropy. These entropies are identical and their difference = O for transitions reflecting maximal entropy, as might be realized in a very long series of fair coin flips in which the entropies = 1. Entropy will be minimal when flipping a two headed coin, here the entropies = 0. More compactly, the non-uniform probabilistic, metric entropy, differing from the maximal topological entropy indicates that the system is in a dynamical state of in-between entropy, written as H7 - Hy + 0.
In the computation of the metric entropy, Hy, the M; is transformed into a transition probability matrix, M;p, called a Markov matrix named for one of the two great Russian mathematicians, both students of Pafnuti Lvovich Chebyshev, the
Markov brothers. The entries of each row in the MM; are transformed into transition
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