LAUSR

As a first step towards modelling real time-series, we study a class of real-variable, bounded processes {Xn,n∈N}$\{X_{n}, n\in \mathbb{N}\}$ defined by a deterministic k$k$-term recurrence relation Xn+k=φ(Xn,…,Xn+k−1)$X_{n+k} = \varphi (X _{n}, \ldots , X_{n+k-1})$. These processes are noise-free. We immerse such a dynamical system into Rk$\mathbb{R}^{k}$ in a slightly distorted way, which allows us to apply the multidimensional techniques introduced by Saussol (Isr. J. Math. 116:223–248, 2000) for deterministic transformations. The hypotheses we need are, most of them, purely analytic and consist in estimates satisfied by the function φ$\varphi $ and by products of its first-order partial derivatives. They ensure that the induced transformation T$T$ is dilating. Under these conditions, T$T$ admits a greatest absolutely continuous invariant measure (ACIM). This implies the existence of an invariant density for Xn$X_{n}$, satisfying integral compatibility conditions. Moreover, if T$T$ is mixing, one obtains the exponential decay of correlations.