Acta Applicandae Mathematicae

, Volume 160, Issue 1, pp 21–34 | Cite as

Exponential Decay of Correlations for a Real-Valued Dynamical System Generated by a \(k\) Dimensional System

  • Lisette JagerEmail author
  • Jules Maes
  • Alain Ninet


As a first step towards modelling real time-series, we study a class of real-variable, bounded processes \(\{X_{n}, n\in \mathbb{N}\}\) defined by a deterministic \(k\)-term recurrence relation \(X_{n+k} = \varphi (X _{n}, \ldots , X_{n+k-1})\). These processes are noise-free. We immerse such a dynamical system into \(\mathbb{R}^{k}\) 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 \(\varphi \) and by products of its first-order partial derivatives. They ensure that the induced transformation \(T\) is dilating. Under these conditions, \(T\) admits a greatest absolutely continuous invariant measure (ACIM). This implies the existence of an invariant density for \(X_{n}\), satisfying integral compatibility conditions. Moreover, if \(T\) is mixing, one obtains the exponential decay of correlations.


ACIM Dynamical systems Decay of correlations Dilating transforms 

Mathematics Subject Classification



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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques, LMR FRE 2011Université de Reims Champagne-ArdenneReimsFrance

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