Annales Henri Poincaré

, Volume 17, Issue 9, pp 2585–2621 | Cite as

On the Convergence to Equilibrium of Unbounded Observables Under a Family of Intermittent Interval Maps

  • Johannes Kautzsch
  • Marc Kesseböhmer
  • Tony Samuel
Article

Abstract

We consider a family \({\{T_{r}: [0, 1] \circlearrowleft \}_{r\in[0, 1]}}\) of Markov interval maps interpolating between the tent map \({T_{0}}\) and the Farey map \({T_{1}}\). Letting \({\mathcal{P}_{r}}\) denote the Perron–Frobenius operator of \({T_{r}}\), we show, for \({\beta \in [0, 1]}\) and \({\alpha \in (0, 1)}\), that the asymptotic behaviour of the iterates of \({\mathcal{P}_{r}}\) applied to observables with a singularity at \({\beta}\) of order \({\alpha}\) is dependent on the structure of the \({\omega}\)-limit set of \({\beta}\) with respect to \({T_{r}}\). The results presented here are some of the first to deal with convergence to equilibrium of observables with singularities.

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Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  • Johannes Kautzsch
    • 1
  • Marc Kesseböhmer
    • 1
  • Tony Samuel
    • 1
    • 2
  1. 1.Fachbereich 3 – Mathematik und InformatikUniversität BremenBremenGermany
  2. 2.Mathematics DepartmentCalifornia Polytechnic State UniversitySan Luis ObispoUSA

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