Volume no :12, Issue no: 1, August (2014)


Author's: Hena Rani Biswas
Pages: [1] - [24]
Received Date: May 9, 2014
Submitted by: Professor Kh. Abdul Maleque


In this paper, we discuss ergodic measure and the various types of mixing for measure-preserving transformations. We discuss the partially understood phenomenon of mixing and indicate some of the contrast between the situations for single measure-preserving transformations. After discussion, we see that strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity. We study Birkhoff ergodic theorem and mixing properties. In this article, we solve some problems of ergodic measure and mixing. We observe that Bernoulli shift is strong mixing. We also show that Markov chain is strong mixing if it is irreducible and aperiodic. Actually, this paper is intended to provide motivation for studying ergodic theory and to describe the major ideas of the subject to a general mathematical audience.


measurable space, probability space, ergodicity, mixing, spectral isomorphism, discrete spectrum.