Hindman, N, Strauss, D and Zamboni, LQ (2018) Recurrence in the dynamical system (X,〈Ts〉s∈S) and ideals of βS. Indagationes Mathematicae, 29 (1). pp. 293312. ISSN 00193577
Abstract
A dynamical system is a pair ( X , 〈 T s 〉 s ∈ S ) , where X is a compact Hausdorff space, S is a semigroup, for each s ∈ S , T s is a continuous function from X to X , and for all s , t ∈ S , T s ∘ T t = T s t . Given a point p ∈ β S , the StoneČech compactification of the discrete space S , T p : X → X is defined by, for x ∈ X , T p ( x ) = p − lim s ∈ S T s ( x ) . We let β S have the operation extending the operation of S such that β S is a right topological semigroup and multiplication on the left by any point of S is continuous. Given p , q ∈ β S , T p ∘ T q = T p q , but T p is usually not continuous. Given a dynamical system ( X , 〈 T s 〉 s ∈ S ) , and a point x ∈ X , we let U ( x ) = p ∈ β S : T p ( x ) is uniformly recurrent . We show that each U ( x ) is a left ideal of β S and for any semigroup we can get a dynamical system with respect to which K ( β S ) = ⋂ x ∈ X U ( x ) and c ℓ K ( β S ) = ⋂ U ( x ) : x ∈ X and U ( x ) is closed . And we show that weak cancellation assumptions guarantee that each such U ( x ) properly contains K ( β S ) and has U ( x ) ∖ c ℓ K ( β S ) ≠ ∅ .
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Copyright, Publisher and Additional Information:  © 2017 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. This is an author produced version of a paper published in Indagationes Mathematicae. Uploaded in accordance with the publisher's selfarchiving policy. 
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Institution:  The University of Leeds 
Academic Units:  The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) 
Depositing User:  Symplectic Publications 
Date Deposited:  22 Jun 2017 10:54 
Last Modified:  05 Jul 2018 03:50 
Status:  Published 
Publisher:  Elsevier 
Identification Number:  https://doi.org/10.1016/j.indag.2016.12.004 
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