#### Date of Submission

2-28-1968

#### Date of Award

2-28-1969

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Ghosh, Jayanta Kumar (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

Ergodic theory is chiefly concerned with the study of transformations on a measure space which preserve the measure. Interesting classes of such transformations are the classes of ergodic, weakly mixing and mixing transformations. The bulk of this thesis is devoted to the study of a closely related family of transformations called the we akly stable transformations; these are more general than the weakly mÃ¤xing transformations.This thesis is divided into three parts. Part I contains preliminary ideas, notations and some results on the invertibility and continuity properties of transition functions (Chapters 1 and 2). In Chapter 3 we collect known results on the splitting theorem and the ergodic theorem in general Banach spaces for later reference.Part II begins with motivating the introduction and study of weakly stable transformations (Chapter 4). In Chapter 5, some simple properties of we akly stable transformations are exhibited. Chapter 6 gives a generalization of the classical mixing theorem for an invertible measure preserving transformation and relates the weak stability of such a transformation to the equality of the invariant o-field in the product space with the product of the invariant o-fields.In Chapter 7, we look at we ak stability from a different angle. Here we introduce we ak stability for semi- groups of contractions on an arbitrary Hilbert space using reversible vectors. This coincides with the previous definition if we consider the group generated by the induced unitary operator in the Lo-space. Generalizations of the results of Chapter 6 are proved for semigroups of contractions in a Hilbert space. As corollaries, we get some known generalizations of the mixing theorem to semigroups of trans- formations. Chapter 8 gives another result on the weak stability of semigroups of measure-preserving transformations relating it to the symmetric invariant sets in the product space.Chapter 9 is concerned with a family of transformations on a probability space endowed with a probability distribution. Associated with the family are a skew product transformation and a transition function. The weak stability properties of these are related to similar properties of the fmaily of transformations.We study automorphisms of compact groups in Chapter 10. Here the we ak stability of an automorphism enables the sub- space of invariant functions in the L, -space to be spanned by the invariant characters. Part II ends with Chapter 11 in which the ergodic decomposition of a transformation is considered. It is proved that if almost all ergodic components are weakly mixing, then the transformation is weakly stable.The problem of existence of invariant measures for families of transformations is considered in Part III. The procedure of using Banach limits for the invariant measure problem for a single transformation is by now well established. The corresponding invariant means for amenable semigroups are utilized to get invariant measures for families of transformations. It is hoped that this will stimulate further research in this direction.

#### Control Number

ISILib-C10527

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Natarajan, S. Dr., "Contribution to the Ergodic Theory." (1969). *Doctoral Theses*. 127.

https://digitalcommons.isical.ac.in/doctoral-theses/127

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842903