Recurrence in Topological Dynamics pp 133-153 | Cite as

# Ellis Semigroups and Ellis Actions

Chapter

## Abstract

A
In terms of the
In general, for a function Φ:
The map Φ is called a
Thus

*semigroup S*is a nonempty set with an associative (usually*not*commutative) multiplication map*M: S*×*S*→*S*. For*p*,*q*∈*S*, we write$$pq = M\left( {p,q} \right) = {M^p}\left( q \right) = {M_q}\left( p \right)$$

(6.1)

*translation maps*, the associative law says$${M^p} \circ {M^q} = {M^{pq}}{M_p} \circ {M_q} = {M_{pq}}$$

(6.2)

*S*×*X*→*X*where*S*is a semigroup, for*p*∈*S*and x ∈*X*, we write$$px = \Phi \left( {p,x} \right) = {\Phi ^p}\left( x \right) = {\Phi _x}\left( p \right)$$

(6.3)

*semigroup action*when for all*p*,*q*∈*S*:$${\Phi ^p} \circ {\Phi ^q} = {\Phi ^{pq}}$$

(6.4)

*M*defines an action of*S*on itself.## Keywords

Ellis Action Compact Space Semi Group Follow Diagram Commute Transitive Point
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1997