Permutation Patterns for Real-Valued Functions

Consider the sequence [x,f(x),f(f(x))=f2(x),…,fn−1(x)] where f is a real-valued function and n≥2. We can associate a permutation to every such sequence by comparing it with x1<x2<...<xn, where xi=fj−1(x) for some j=1,2,…,n. Permutations that arise from these sequences are called allowed permutations and those that do not are called forbidden permutations.For example, the logistic map, f:[0,1]→[0,1] is defined by f(x)=rx(1−x) where 0≤r≤4, for any x. We focus on enumerating the number of forbidden permutations for the logistic map and other functions, including trigonometric functions. For example, for the n=3 case, we have found that the one-line permutation (321) is a forbidden permutation for the function sin(πx).

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