The following is a concise exposition on the conjecture and three of its proofs for the case of positive entropy, by D. Rudolph [18] , by B. Host [13] and by W. Parry [17]. A simpler theorem of R. Lyons [15] -preceding them -is also presented and proved. This is a survey, no new results are introduced.

The notion of Bohr chaos was introduced in [3,4]. We answer a question raised in [3] of whether a non uniquely ergodic minimal system of positive topological entropy can be Bohr chaotic. We also prove that all systems with the specification property are Bohr chaotic, and by this line of thought give an independent proof (and stengthening) of theorem 1 of [3] for the case of invertible systems. In addition, we present an obstruction for Bohr chaos: a system with fewer than a continuum of ergodic invariant probability measures cannot be Bohr chaotic.

We define basic notions in the category of conic representations of a topological group and prove elementary facts about them. We show that a conic representation determines an ordinary dynamical system of the group together with a multiplier, establishing facts and formulae connecting the two categories. The topic is also closely related to the affine representations of the group. The central goal was attaining a better understanding of irreducible conic representations of a group, and -particularly -to determine whether there is a phenomenon analogous to the existence of a universal irreducible affine representation of a group in our category (the general answer is negative). Then we inspect embeddings of irreducible conic representations of semi-simple Lie groups in some "regular" conic representation they possess. We conclude with what is known to us about the irreducible conic representations of SL2 (R).

No abstract