The invariant density for a class of discrete‐time maps involving an arbitrary monotonic function operator and an integer parameter

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Abstract
When xt and xt+1 represent two random variables, each belonging to a real interval [0,1] and being related by a first‐order difference equation xt+1=F(xt), called a discrete‐time map, the probability density distribution connected with xt can be translated into that associated with xt+1. This yields an evolution equation by means of which one can construct an infinite sequence {wt(x)‖t∈N, x∈[0,1]} starting from an integrable function w0(x) normalized to unity on [0,1]. The question of the convergence of the sequence toward a so‐called invariant density function w(x) as t→+∞ and the problem of finding this limit were examined by a number of authors, mostly studying isolated cases. The present paper solves the problem for a class of discrete‐time maps characterized by xt+1 =f(‖sn[l sn−1f−1(xt)]‖), l∈{2,3,...}, whereby f is a real, continuous, monotonically increasing function mapping [0,1] onto itself and sn is the usual symbol for the sinelike Jacobian elliptic function with modulus k∈[0,1] (including the sine function). Convergence is proven under very general conditions on w0(x) and an explicit formula to calculate w(x) is established. Some properties of w(x) are discussed. A necessary and sufficient condition for the symmetry of w(x) about x= 1/2 is obtained and attention is also devoted to the inverse problem, leading to a reformulation of the discrete‐time map of the type cited above which corresponds to a given invariant density. The examples of practical application considered here cover almost all special cases which were treated in the literature thus far, as well as new cases.

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