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Where can I find a proof that a distal continuous function of a compact metric space is surjective?

PS: The person asking the question Is there an elementary proof that distal maps are invertible? says he knows two proofs that use the enveloping semigroup, and the Stone-Čech compactification. I am interested in the references where I can find those proofs

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    $\begingroup$ Does this answer your question? mathoverflow.net/questions/393648/… $\endgroup$
    – Joseph Van Name
    Commented May 15, 2022 at 18:47
  • $\begingroup$ The person asking the question says he knows two proofs that use the enveloping semigroup, and the Stone-Čech compactification. I am interested in the references where I can find those proofs. $\endgroup$
    – J.C.
    Commented May 15, 2022 at 19:05
  • $\begingroup$ Then I have added a "reference-request" tag. $\endgroup$
    – Joseph Van Name
    Commented May 15, 2022 at 19:53
  • $\begingroup$ Maybe look here arxiv.org/pdf/1708.00996.pdf#page11 $\endgroup$
    – Benjamin Steinberg
    Commented May 16, 2022 at 2:40

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OP here! The Stone Cech proof can be found on page 33 of Bergelson’s survey on Ergodic Ramsey theory.

The enveloping semigroup proof can be found on page 60 of Brown’s book Topological Dynamics and Ergodic Theory.

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  • $\begingroup$ Thank you, indeed i found the proofs in those references, furthermore, in the Bergelson’s survey the author refers to the article "Ebrahim Salehi and David B. Ellis. Problems and Solutions: Solutions: Revivals: 6612. Amer. Math. Monthly". Where we can found the enveloping semigroup proof . $\endgroup$
    – J.C.
    Commented May 25, 2022 at 23:14

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