Characterizations of regular almost periodicity in compact minimal abelian flows
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- by Alica Miller and Joseph Rosenblatt PDF
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Abstract:
Regular almost periodicity in compact minimal abelian flows was characterized for the case of discrete acting group by W. Gottschalk and G. Hedlund and for the case of $0$-dimensional phase space by W. Gottschalk a few decades ago. In 1995 J. Egawa gave characterizations for the case when the acting group is $\mathbb {R}$. We extend Egawa’s results to the case of an arbitrary abelian acting group and a not necessarily metrizable phase space. We then show how our statements imply previously known characterizations in each of the three special cases and give various other applications (characterization of regularly almost periodic functions on arbitrary abelian topological groups, classification of uniformly regularly almost periodic compact minimal $\mathbb {Z}$- and $\mathbb {R}$-flows, conditions equivalent with uniform regular almost periodicity, etc.).References
- Joseph Auslander, Endomorphisms of minimal sets, Duke Math. J. 30 (1963), 605–614. MR 155311
- Joseph Auslander, Minimal flows and their extensions, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 122. MR 956049
- Joseph Auslander and Marianne Guerin, Regional proximality and the prolongation, Forum Math. 9 (1997), no. 6, 761–774. MR 1480556, DOI 10.1515/form.1997.9.761
- Jozeph Auslander and Frank Hahn, Point transitive flows, algebras of functions and the Bebutov system, Fund. Math. 60 (1967), 117–137. MR 221489, DOI 10.4064/fm-60-2-117-137
- A. Besicovitch, Almost periodic functions, Cambridge University Press, Cambridge, 1932.
- C. Corduneanu, Almost periodic functions, Chelsea Publishing Company, New York, N.Y., 1989.
- Jir\B{o} Egawa, Eigenvalues of some almost automorphic functions, Kobe J. Math. 2 (1985), no. 2, 149–161. MR 847182
- Jir\B{o} Egawa, Eigenvalues of some almost periodic functions, Proc. Amer. Math. Soc. 115 (1992), no. 2, 535–540. MR 1079890, DOI 10.1090/S0002-9939-1992-1079890-3
- Jir\B{o} Egawa, A characterization of regularly almost periodic minimal flows, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), no. 10, 225–228 (1996). MR 1373387
- Jir\B{o} Egawa, Eigenvalues of some distal functions, Proc. Amer. Math. Soc. 126 (1998), no. 1, 273–278. MR 1458868, DOI 10.1090/S0002-9939-98-04488-8
- Robert Ellis, Distal transformation groups, Pacific J. Math. 8 (1958), 401–405. MR 101283, DOI 10.2140/pjm.1958.8.401
- Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
- Robert Ellis and W. H. Gottschalk, Homomorphisms of transformation groups, Trans. Amer. Math. Soc. 94 (1960), 258–271. MR 123635, DOI 10.1090/S0002-9947-1960-0123635-1
- Robert Ellis and Harvey Keynes, A characterization of the equicontinuous structure relation, Trans. Amer. Math. Soc. 161 (1971), 171–183. MR 282357, DOI 10.1090/S0002-9947-1971-0282357-4
- A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR 0460799, DOI 10.1007/BFb0070324
- S. Glasner and D. Maon, Rigidity in topological dynamics, Ergodic Theory Dynam. Systems 9 (1989), no. 2, 309–320. MR 1007412, DOI 10.1017/S0143385700004983
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- W. H. Gottschalk, Substitution minimal sets, Trans. Amer. Math. Soc. 109 (1963), 467–491. MR 190915, DOI 10.1090/S0002-9947-1963-0190915-6
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- E. Hewitt and K. Ross, Abstract harmonic analysis, Springer-Verlag, Berlin, Heidelberg, New York, 1963.
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- D. McMahon and T. S. Wu, On weak mixing and local almost periodicity, Duke Math. J. 39 (1972), 333–343. MR 336724, DOI 10.1215/S0012-7094-72-03940-3
- D. McMahon and T. S. Wu, On the connectedness of homomorphisms in topological dynamics, Trans. Amer. Math. Soc. 217 (1976), 257–270. MR 413067, DOI 10.1090/S0002-9947-1976-0413067-2
- D. McMahon and T. S. Wu, Notes on topological dynamics. IV. Relative equicontinuity and its variations, Bull. Inst. Math. Acad. Sinica 8 (1980), no. 2-3, 277–281. MR 595535
- Wenxian Shen and Yingfei Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc. 136 (1998), no. 647, x+93. MR 1445493, DOI 10.1090/memo/0647
- W. A. Veech, Almost automorphic functions on groups, Amer. J. Math. 87 (1965), 719–751. MR 187014, DOI 10.2307/2373071
- William A. Veech, The equicontinuous structure relation for minimal Abelian transformation groups, Amer. J. Math. 90 (1968), 723–732. MR 232377, DOI 10.2307/2373480
- J. de Vries, Elements of topological dynamics, Mathematics and its Applications, vol. 257, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1249063, DOI 10.1007/978-94-015-8171-4
- A. Weil, L’ Integration dans les Groupes Topologiques. Hermann, Paris, 1940.
Additional Information
- Alica Miller
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
- Email: amiller@math.uiuc.edu
- Joseph Rosenblatt
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
- MR Author ID: 150595
- Email: jrsnbltt@math.uiuc.edu
- Received by editor(s): November 3, 2002
- Received by editor(s) in revised form: June 19, 2003
- Published electronically: February 4, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 4909-4929
- MSC (2000): Primary 37B05, 43A60; Secondary 43A40, 54H20
- DOI: https://doi.org/10.1090/S0002-9947-04-03538-X
- MathSciNet review: 2084405