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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
Article

Abstract

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.

Keywords

ACIM Dynamical systems Decay of correlations Dilating transforms 

Mathematics Subject Classification

37C40 

References

  1. 1.
    Alves, J.F., Freitas, J.M., Luzzatto, S., Vaienti, S.: From rates of mixing to recurrence times via large deviations. Adv. Math. 228(2), 1203–1236 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Gouëzel, S.: Sharp polynomial estimates for the decay of correlations. Isr. J. Math. 139, 29–65 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180, 119–140 (1982) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ionescu Tulcea, C.T., Marinescu, G.: Théorie ergodique pour des classes d’opérations non complètement continues. Ann. Math. 52(2), 140–147 (1950) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jager, L., Maes, J., Ninet, A.: Exponential decay of correlations for a real-valued dynamical system embedded in \(\mathbb{R}^{2}\). C. R. Math. 353(11), 1041–1045 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lasota, A., Mackey, M.C.: Chaos, Fractals and Noise: Stochastic Aspects of Dynamics. Springer, New York (1998) zbMATHGoogle Scholar
  7. 7.
    Li, T.Y., Yorke, J.A.: Ergodic maps on \([0,1]\) and nonlinear pseudo-random number generators. Nonlinear Anal., Theory Methods Appl. 2(4), 473–481 (1978) CrossRefzbMATHGoogle Scholar
  8. 8.
    Sarig, O.: Subexponential decay of correlations. Invent. Math. 150, 629–653 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Saussol, B.: Absolutely continuous invariant measures for multidimensional expanding maps. Isr. J. Math. 116, 223–248 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Young, L.S.: Recurrence times and rates of mixing. Isr. J. Math. 110, 153–188 (1999) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© 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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