Operator renewal theory and mixing rates for dynamical systems with infinite measure

• 成果类型: Article
• 署名作者: Melbourne, Ian; Terhesiu, Dalia
• 署名单位: University of Surrey
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-011-0361-4
• 发表日期: 2012
• 页码: 61-110
• 关键词: perron-frobenius operator; interval maps; LIMIT-THEOREMS; decay; transformations

摘要

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates L (n) of the transfer operator. This was previously an intractable problem. Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points. In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of ) for the class of systems under consideration. In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for L (n) and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.