#### Date of Submission

1-29-2000

#### Date of Award

1-29-2001

#### 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

Dani, S. G.

#### Abstract (Summary of the Work)

The main theme of this thesis is topological classiication of aline Hows on homogeneous spaces and rigidity of equivariant continuous maps between such dows. Both these aspects have been extensively studied in the literature for subgroup actions (cf. [Be), [B-Dj and [Wi) and for automorphism flows of the cyclic group (cf. [Ar]. [K-R] and [C-S). We will consider similar questions in more general situations. A detailed outline is given below. For a topological group T, by a r-flow we mean pair (X,p), where X is a opological space and p is a continuous action of I on X. For any two r-flows X. p) and (X', a), a continuous map f : X + X' is said to be r-equivariant /o pl1) = o(1) o f. Vy â‚¬ T. Two I-flows (X, p) and (.X",o) are said to be :pologically conjugate if there exists a F-equivariant homeomorphism f : Xâ†’ X. and they are said to be orbit equiralent if there exists a homeomorphism ::X - X" which takes orbits under p to orbits under o. If G is a locally compact topological group and HCG is a closed subgroup. ben the quotient space .V = G/H is called a homogeneous space. If x1, = G1/H1 apd x2, = G2/H2, are homogeneous spaces then a continuous map from X, to X2 s said to be a homomorphism if it is induced by a continuous homomorphism m G1; to G2 which maps H1, into H2. Isomorphisms and automorphisms of ho- sozeneous spaces are defined similarly. A continuous map f from X1 to X2, is said to be affine if there exists an element go of G2 and a continuous homomorphism 0 from G1, to G2 such that 0(H1,) C H2, and f(gH1;) = go0(g)H2 Vg â‚¬ G1,. A r-flow (V. p) is said to be affine if for all in r. p() is an affine map. (X, p) is said to be an automorphism flow (resp. a translation flow ) if each p(y), y er, is an automorphism of X (resp. a translation on V). Two T-flows (X1,P) and (X2,0) are said to be algebraically conjugate if there exists a r-equivariant isomorphism from X1 to X2. In Chapter 1 we set up various definitions, notations etc. and discuss the background material. In Chapter 2 we consider flows on compact connected metrizable abelian groups. We give various sufficient conditions for 'rigidity' of T-equivariant continuous maps in this situation. In particular we prove that if G and H are compact connected metrizable abelian groups and p, o are affine actions of a discrete group r on G and H respectively such that (H,o) is ex- pansive then every r-equivariant continuous mapf: (G, p) + (H, o) is an affine map. We also classify certain classes of translation flows on such groups up to orbit equivalence and topological conjngacy. For r =R, we prove that two one- parameter translation flows on such groups are orbit equivalent if and only if they are algebraically conjugate after a change of scale. This generalizes an earlier result in the case of tori (see [Bel). Also for any topological group r, we prove that two translation flows of F on compact connected metrizable abelian groups are topologically conjugate if and only if they are algebraically conjugate.

#### Control Number

ISILib-TH201

#### 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

Bhattacharya, Siddhartha Dr., "Topological Conjugacy and Rigidity of Affine Actions." (2001). *Doctoral Theses*. 332.

https://digitalcommons.isical.ac.in/doctoral-theses/332

## 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:28843422