Consider an absolutely continuous probability measure \(\mu\) on the circle \(\mathbb{T}^1\), and a \(\mu\)-preserving continuous, piecewise monotone, absolutely continuous and mixing map \(f: \mathbb{T}^1\to \mathbb{T}^1\). Consider also a continuous map \(\varphi: \mathbb{T}^1\to\mathbb{T}^1\), and define the continuous function \(x: \mathbb{R}_+\to \mathbb{T}^1\) by: \(x:= \varphi\) on \([0,1[\), and \(x(t):= f^n\circ\varphi(t- n)\) for \(n\in\mathbb{N}^*\), \(n\leq t< n+1\). Then, under some technical additinal assumptions, the author establishes in articular the finite-dimensional marginals \((x(T+ t_1),\dots, x(T+ t_n))\), for \(t_1<\cdots< t_n\), converge to \(\mu^{\otimes n}\) as \(T\to\infty\).
This exhibits an asymptotic behaviour of the deterministic function \(x\) of chaotic-stochastic type. A somewhat more general result is actually proved.