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Central limit theorem behavior in the skew tent map

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  • Mackey, Michael C.
  • Tyran-Kamińska, Marta

Abstract

In this paper we study and establish central limit theorem behavior in the skew (generalized) tent map transformation T: Y→Y originally considered by Billings and Bollt [Billings L, Bollt EM. Probability density functions of some skew tent maps. Chaos, Solitons & Fractals 2001; 12: 365–376] and Ito et al. [Ito S, Tanaka S, Nakada H. On unimodal linear transformations and chaos. II. Tokyo J Math 1979; 2: 241–59]. When the measure ν is invariant under T, the transfer operator PT:L1(ν)→L1(ν) governing the evolution of densities f under the action of the skew tent map, as well as the unique stationary density, are given explicitly for specific transformation parameters. Then, using this development, we solve the Poisson equation f=PTf+ϕ for two specific integrable observables ϕ and explicitly calculate the variance σ(ϕ)2=∫Yϕ2(y)ν(dy).

Suggested Citation

  • Mackey, Michael C. & Tyran-Kamińska, Marta, 2008. "Central limit theorem behavior in the skew tent map," Chaos, Solitons & Fractals, Elsevier, vol. 38(3), pages 789-805.
    • Handle: RePEc:eee:chsofr:v:38:y:2008:i:3:p:789-805
    DOI: 10.1016/j.chaos.2007.01.013
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    Cited by:

    1. Freitas, Ana Cristina Moreira, 2009. "Statistics of the maximum for the tent map," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 604-608.

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