×

zbMATH — the first resource for mathematics

A bound for the rate of decrease of correlation in one-dimensional dynamical systems. (English. Russian original) Zbl 0549.28024
Funct. Anal. Appl. 18, 50-52 (1984); translation from Funkts. Anal. Prilozh. 18, No. 1, 61-62 (1984).
Let T be a mapping of [0,1] into itself such that there exists a finite partition \(0=a_ 0<a_ 1<...<a_ k=1\) for which T is monotone and of class \(C^ 1\) on each \((a_{i-1},a_ i)\), the function 1/\(| T'|\) is of bounded variation on each \([a_{i-1},a_ i]\) and \(\inf_{x}| T'(x)| >1.\) Then T is known to have an invariant measure \(\mu\) absolutely continuous with respect to Lebesgue measure. For \(f,g\in C^ 1[0,1]\) the n-th correlation is defined by \[ K_ n(f,g)=\int f(T^ nx)g(x)d\mu -\int f(x)d\mu\int g(x)d\mu. \] The author investigates the rate of decrease of \(K_ n(f,g)\) for mixings T.
Reviewer: V.V.Peller

MSC:
28D10 One-parameter continuous families of measure-preserving transformations
37A99 Ergodic theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. Wong, Trans. Am. Math. Soc.,246, 493-500 (1978). · doi:10.1090/S0002-9947-1978-0515555-9
[2] Ya. G. Sinai, ”Some precise results about the decrease of correlation,” Novosibirsk: Preprint of the Institute of Nuclear Physics, Siberian Branch, Academy of Sciences of the USSR (1968).
[3] F. Hofbauer and G. Keller, Math. Z.,180, 119-140 (1982). · Zbl 0485.28016 · doi:10.1007/BF01215004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.