By David B. Ellis, Robert Ellis

Targeting the function that automorphisms and equivalence kinfolk play within the algebraic concept of minimum units offers an unique remedy of a few key points of summary topological dynamics. Such an procedure is gifted during this lucid and self-contained ebook, resulting in less complicated proofs of classical effects, in addition to supplying motivation for additional research. minimum flows on compact Hausdorff areas are studied as icers at the common minimum circulate M. the gang of the icer representing a minimum circulation is outlined as a subgroup of the automorphism crew G of M, and icers are built explicitly as relative items utilizing subgroups of G. Many classical effects are then acquired through analyzing the constitution of the icers on M, together with an evidence of the Furstenberg constitution theorem for distal extensions. This e-book is designed as either a consultant for graduate scholars, and a resource of attention-grabbing new principles for researchers.

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**Example text**

U = ηuv = ηv. 10 6 12. η ∈ Kη ∩ Kv. 13. η = v. 12 we see that any two minimal ideals in E(X, T ) are isomorphic as minimal flows in a natural way. 15 Let: (i) (ii) (iii) (iv) (v) (X, T ) be a flow, E = E(X, T ), I, K ⊂ E be minimal ideals in E, u2 = u ∈ I be an idempotent, and v 2 = v ∈ K with u ∼ v. Then the map Lv : (I, T ) p the map Lu . PROOF: → → (K, T ) is an isomorphism, its inverse being vp We leave the proof as an exercise for the reader. The structure of the minimal ideals in the enveloping semigroup E(X, T ) of any flow (X, T ) described above, and the minimal idempotents themselves play an important role (see in particular section 4) in the study of the dynamics of (X, T ).

Despite this, some deep results along these lines can be obtained by deducing the general case from metric considerations. 24 that every flow which is both topologically transitive and distal, must be minimal. This result has the general case of the Furstenberg theorem as an immediate consequence. This result can also be generalized to the case of homomorphisms of minimal flows. Here one can prove that a homomorphism of minimal flows which is both weak mixing (so that the corresponding equivalence relation is topologically transitive) and distal must be trivial.

Conversely given any flow (X, T ), the set {π t | t ∈ T } is a subgroup of XX consisting of continuous maps. In this case we obtain a group homomorphism of T into a subgroup of XX , allowing the following definition of the enveloping semigroup of the flow (X, T ). 8 Let (X, T ) be a flow. 1 the map T t X tinuous extension X : βT → X . The image of X , X (βT ) → X X has a con→ πt = {π t | t ∈ T } ≡ E(X, T ) is clearly a subsemigroup of XX , which we call the enveloping semigroup of the flow (X, T ) and denote by E(X, T ) or simply E(X).