Limit theorems for wobbly interval intermittent maps. (English) Zbl 1482.60035
The authors consider the following interval maps \[ f_{M_j}(x)=\left\{\begin{array}{lr} x(1+M_j(x)x^\alpha),& \ x\in[0,1/2],\\
2x-1,& \ x\in(1/2,1], \end{array}\right. \] where \({\alpha\in(0,1)}\) and \[ M_1(x)=\frac{c_0}{2^{\{c_1^{-1}\log x\}}},\ \ M_2(x)=a\left(1+b\sin\left(\frac{2\pi}{c_2}\log x\right)\right) \] for certain constants \({a, b, c_0, c_1, c_2}.\)
The authors obtain limit laws for such maps and Hölder observables. In turns out, that the laws are similar to some classical semistable laws (see [Z. Megyesi, Acta Sci. Math. 66, No. 1–2, 403–434 (2000; Zbl 0957.60020)]). The main theorem is partially motivated by the results of third author and P. Kevei [J. Theor. Probab. 33, No. 4, 2027–2060 (2020; Zbl 1454.37015)] and as the authors believe it can be modified for infinite measure case \({\alpha\geq1}.\)
The authors obtain limit laws for such maps and Hölder observables. In turns out, that the laws are similar to some classical semistable laws (see [Z. Megyesi, Acta Sci. Math. 66, No. 1–2, 403–434 (2000; Zbl 0957.60020)]). The main theorem is partially motivated by the results of third author and P. Kevei [J. Theor. Probab. 33, No. 4, 2027–2060 (2020; Zbl 1454.37015)] and as the authors believe it can be modified for infinite measure case \({\alpha\geq1}.\)
Reviewer: Ivan Podvigin (Novosibirsk)
MSC:
| 60F05 | Central limit and other weak theorems |
| 37A50 | Dynamical systems and their relations with probability theory and stochastic processes |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
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