Return time statistics for unimodal maps
نویسندگان
چکیده
منابع مشابه
Return Time Statistics for Invariant Measures for Interval Maps with Positive Lyapunov Exponent
We prove that multimodal maps with an absolutely continuous invariant measure have exponential return time statistics around a.e. point. We also show a ‘polynomial Gibbs property’ for these systems, and that the convergence to the entropy in the Ornstein-Weiss formula has normal fluctuations. These results are also proved for equilibrium states of some Hölder potentials.
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We prove that return time statistics of a dynamical system do not change if one passes to an induced (i.e. rst return) map. We apply this to show exponential return time statistics in i) smooth interval maps with nowhere{dense critical orbits and ii) certain interval maps with neutral xed points. The method also applies to iii) certain quadratic maps of the complex plane.
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For non markovian, piecewise monotonic maps of the interval associated to a potential, we prove that the law of the entrance time in a cylinder, when renormalized by the measure of the cylinder, converges to an exponential law for almost all cylinders. Thanks to this result, we prove that the fluctuations of Rn, first return time in a cylinder, are lognormal.
متن کاملReturn Time Statistics of Invariant Measures for Interval Maps with Positive Lyapunov Exponent
We prove that multimodal maps with an absolutely continuous invariant measure have exponential return time statistics around almost every point. We also show a ‘polynomial Gibbs property’ for these systems, and that the convergence to the entropy in the Ornstein-Weiss formula has normal fluctuations. These results are also proved for equilibrium states of some Hölder potentials.
متن کاملBifurcations of Unimodal Maps
We review recent results that lead to a very precise understanding of the dynamics of typical unimodal maps from the statistical point of view. We also describe the (generalized) renormalization approach to the study of the statistical properties of typical unimodal maps.
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ژورنال
عنوان ژورنال: Fundamenta Mathematicae
سال: 2003
ISSN: 0016-2736,1730-6329
DOI: 10.4064/fm176-1-6