The original definition was introduced by Adler, Konheim and McAndrew in 1965. 8.5 Topological entropy for nonautonomous dynamical systems. 8 Generalizations of topological entropy.7.1 Topological tail entropy and symbolic extension entropy.6 Topological entropy in some special cases.5 Relation with Kolmogorov-Sinai entropy.4 Basic properties of topological entropy.In what follows \(\log\) denotes \(\log_2\) (although this choice is arbitrary). Roughly, it measures the exponential growth rate of the number of distinguishable orbits as time advances. Topological entropy is a nonnegative number which measures the complexity of the system. Let \((X,T)\) be a topological dynamical system, i.e., let \(X\) be a nonempty compact Hausdorff space and \(T:X\to X\) a continuous map. The number of orbits distinguishable in \(n\) steps grows as \(2^n\ ,\) generating the topological entropy \((1/n)\log_2(2^n) = 1\. Similarly, there are eight points (the black points), whose orbits are similarly distinguished in three steps (after one iterate the black points become the red and yellow points, after another iterate they become the blue, violet and green points). But there exist already four different points whose orbits can be distinguished in two steps: the red points are mapped onto the blue and violet points and any two of them are distinguished either immediately or after applying the transformation once. Initially there are at most two distinguishable points, for example, the blue points. Suppose that only points that are in opposite halves of the rectangle can be distinguished. Figure 1: Topological entropy generated in a so-called horseshoe: the rectangle is stretched, bent upward and placed over itself.
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