Law of iterated logarithm and invariance principle for one-parameter families of interval maps

• 成果类型: Article
• 署名作者: Schnellmann, Daniel
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite PSL; Ecole Normale Superieure (ENS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0575-7
• 发表日期: 2015
• 页码: 365-409
• 关键词: piecewise monotonic transformations; unimodal maps; expanding maps; ergodic-theory; systems; CONVERGENCE; DYNAMICS; numbers; point

摘要

We show that for almost every map in a transversal one-parameter family of piecewise expanding unimodal maps the Birkhoff sum of suitable observables along the forward orbit of the turning point satisfies the law of iterated logarithm. This result will follow from an almost sure invariance principle for the Birkhoff sum, as a function on the parameter space. Furthermore, we obtain a similar result for general one-parameter families of piecewise expanding maps on the interval.