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On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps. (English) Zbl 1359.82017

The interval maps with intermittent-like behaviour on the unit interval \([0,1]\) have been widely studied during the last years with the help of the associated transfer operators which allow to solve problems concerning the statistical behaviour of these maps. A lot of effort has been done on different possible applications except on an unstable fixed point at which trajectories are considerably slowed down. This situation is caused by the fact that at this point we have an interplay of chaotic and regular dynamics – standard methods of ergodic theory cannot be applied. In the paper, the authors focus on a family \(T_r\), \(r\in[0,1]\), of Markov interval maps interpolating between the tent map \(T_0\) and the Farey map \(T_1\). For \(r\in[0,1)\) a lot of properties are given with detailed descriptions and analyses presented in the literature. But the case \(r=1\) requires more sophisticated approaches. Necessary details for understanding the authors’ approach are presented in Section 1, while in Section 2 we have the most important definitions needed for the understanding of the proposed solutions. Section 3 gives the obtained formal results in order to present further details in Section 4. In this section, the most important part is devoted to considerations about the case \(r=1\). Section 5 presents necessary proofs for the obtained solutions and proposals.

MSC:

82C35 Irreversible thermodynamics, including Onsager-Machlup theory
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
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