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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

28D10 One-parameter continuous families of measure-preserving transformations
37A99 Ergodic theory
Full Text: DOI
[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
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