Abstract
We prove that for every countable discrete group G, there is a G-flow on ω* that has every G-flow of weight ≤ ℵ1 as a quotient. It follows that, under the Continuum Hypothesis, there is a universal G-flow of weight \(\leq \mathfrak{c}\).
Applying Stone duality, we deduce that, under CH, there is a trivial automorphism τ of P(ω)/fin with every other automorphism embedded in it, which means that every other automorphism of P(ω)/fin can be written as the restriction of τ to a suitably chosen subalgebra. We give an exact characterization of all trivial automorphisms with this property.
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References
E. Akin, The General Topology of Dynamical Systems, Graduate Studies in Mathematics, Vol. 1, American Mathematical Society, Providence, RI, 2010.
I. Bandlow, A construction in set-theoretic topology by means of elementary substructures, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 37 (1991), 467–480.
A. Błaszczyk and A. Szymański, Concerning Parovičenko’s theorem, L’Académie Polonaise des Sciences. Bulletin. Série des Sciences Mathématiques 28 (1980), 311–314.
R. Bowen, ω-limit sets for axiom A diffeomorphisms, Journal of Differential Equations 18 (1975), 333–339.
W. R. Brian, P-sets and minimal right ideals in ℕ*, Fundamenta Mathematicae 229 (2015), 277–293.
W. R. Brian, Abstract omega-limit sets, Journal of Symbolic Logic 83 (2018), 477–495.
A. Dow and K. P. Hart, A universal continuum of weight ℵ, Transactions of the American Mathematical Society 353 (2000), 1819–1838.
I. Farah, Analytic quotients: theory of lifting for quotients over analytic ideals on the integers, Memoirs of the American Mathematical Society 148 (2000).
S. Geschke, The shift on P(ω)/fin, unpublished manuscript, available at http://www.math.uni-hamburg.de/home/geschke/publikationen.html.en.
N. Hindman and D. Strauss, Algebra in the Stone–Čech Compactification, Walter de Gruyter, Berlin, 2012.
P. T. Johnstone, Stone Spaces, Cambridge Studies in Advanced Mathematics, Vol. 3, Cambridge University Press, Cambridge, 1982.
J. van Mill, An introduction to βω, in Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, pp. 503–560.
N. Noble and M. Ulmer, Quotienting functions on Cartesian products, Transactions of the American Mathematical Society 163 (1972), 329–339.
I. I. Parovičenko, A universal bicompact of weight ℵ1, Soviet Mathematics Doklady 4 (1963), 592–595.
E. V. Shchepin, Real functions and canonical sets in Tikhonov products and topological groups, Russian Mathematical Surveys 31 (1976), 17–27.
S. Shelah, Proper Forcing, Lecture Notes in Mathematics, Vol. 940, Springer, Berlin, 1982.
S. Shelah and J. Steprāns, PFA implies all automorphisms are trivial, Proceedings of the American Mathematical Society 104 (1988), 1220–1225.
S. Todorčević, Partition Problems in Topology, Contemporary Mathematics, Vol. 84, American Mathematical Society, Providence, RI, 1989.
B. Velickovic, OCA and automorphisms of P(ω)/fin, Topology and its Applications 49 (1992), 1–12.
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Brian, W. Universal flows and automorphisms of P(ω)/fin. Isr. J. Math. 233, 453–500 (2019). https://doi.org/10.1007/s11856-019-1913-3
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DOI: https://doi.org/10.1007/s11856-019-1913-3