On $sncc$-Inheritance of Pointwise Almost Periodicity in Flows
Keywords:
almost periodic point, flow, inheritanceAbstract
Let $H$ be a subnormal co-compact closed subgroup of a Hausdorff topological group $T$ and $X$ a compact Hausdorff space. We prove the inheritance theorem: A point of $X$ is almost periodic (a.p.) for $T \curvearrowright X$ if and only if it is a.p. for $H \curvearrowright X$. Moreover, if $T \curvearrowright X$ is minimal with $H \triangleleft T$, then $\mathscr{O}_H: X \rightarrow 2^X, x \mapsto \overline{H x}$ is a continuous mapping, and $T \curvearrowright X / H$ is an a.p. nontrivial factor of $T \curvearrowright X$ if and only if $T \curvearrowright X \times T / H$ is not minimal.
References
Joseph Auslander, Minimal Flows and Their Extensions. North-Holland Mathematics Studies, 153. Amsterdam: North-Holland, 1988.
L. Auslander, L. Green, and F. Hahn, Flows on Homogeneous Spaces. With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. Annals of Mathematics Studies, No. 53. Princeton, N.J.: Princeton University Press, 1963.
I. U. Bronšteǐn, Extensions of Minimal Transformation Groups. The Hague: Martinus Nijhoff Publishers, 1979.
Xiongping Dai, On recurrence in zero-dimensional locally compact flow with compactly generated phase group, Illinois J. Math. 66 (2022), 595-625.
J. de Vries, Elements of Topological Dynamics. Dordrecht: Kluwer Academic Publishers Group, 1993.
Robert Ellis, Lectures on Topological Dynamics. New York: W.A. Benjamin, 1969.
Shmuel Glasner, Proximal Flows. Lecture Notes in Mathematics, 517. Berlin-New York: Springer-Verlag, 1976.
W. H. Gottschalk, Almost periodic points with respect to transformation semigroups, Annals of Math.(2) 47 (1946), 762-766.
Walter Helbig Gottschalk and Gustav Arnold Hedlund, Topological Dynamics. American Mathematical Society Colloquium Publications, Vol. 36. Providence, RI: American Mathematical Society, 1955.
Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis. Vol. I: Structure of Topological Groups, Integration Theory, Group Representations. New York, Berlin: Springer-Verlag, 1963.
Harvey B. Keynes, A study of the proximal relation in coset transformation groups, Trans. Amer. Math. Soc. 128 (1967), 389-402.
V.V. Nemyskii and V.V. Stepanov, Qualitative Theory of Differential Equations. Princeton, N.J.: Princeton Univ. Press, 1960.
William A. Veech, Topological dynamics, Bull. Amer. Math. Soc. 83 (1977), no. 5, 775-830.
