On the connectedness of homomorphisms in topological dynamics

Authors:
D. McMahon and T. S. Wu

Journal:
Trans. Amer. Math. Soc. **217** (1976), 257-270

MSC:
Primary 54H20

DOI:
https://doi.org/10.1090/S0002-9947-1976-0413067-2

MathSciNet review:
0413067

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let (*X, T*) be a minimal transformation group with compact Hausdorff phase space. We show that if is a distal homomorphism and has a structure similar to the structure Furstenberg derived for distal minimal sets, then for *T* belonging to a class of topological groups T, the homomorphism has connected fibers, where is the relativized equicontinuous structure relation. The class T is defined by Sacker and Sell as consisting of all groups *T* with the property that there is a compact set such that *T* is generated by each open neighborhood of *K*. They show that for such *T*, a distal minimal set which is a finite-to-one extension of an almost periodic minimal set is itself an almost periodic minimal set. We provide an example that shows that the restriction on *T* cannot be dropped.

As one of the preliminaries to the above we show that given , the relation induced by the components in the fibers relative to , i.e., if and only if *x* and are in the same component of , is a closed invariant equivalence relation.

We also consider the question of when a minimal set (*X, T*) is such that is finite for some *x* in *X*, where *Q* is the regionally proximal relation. This problem was motivated by Veech's work on almost automorphic minimal sets, i.e., the case in which is a singleton for some *x* in *X*.

**[1]**Robert Ellis,*Distal transformation groups*, Pacific J. Math.**8**(1958), 401–405. MR**0101283****[2]**Robert Ellis,*A semigroup associated with a transformation group*, Trans. Amer. Math. Soc.**94**(1960), 272–281. MR**0123636**, https://doi.org/10.1090/S0002-9947-1960-0123636-3**[3]**Robert Ellis,*The construction of minimal discrete flows*, Amer. J. Math.**87**(1965), 564–574. MR**0185589**, https://doi.org/10.2307/2373063**[4]**Robert Ellis,*Group-like extensions of minimal sets*, Trans. Amer. Math. Soc.**127**(1967), 125–135. MR**0221492**, https://doi.org/10.1090/S0002-9947-1967-0221492-2**[5]**Robert Ellis,*The structure of group-like extensions of minimal sets*, Trans. Amer. Math. Soc.**134**(1968), 261–287. MR**0238293**, https://doi.org/10.1090/S0002-9947-1968-0238293-2**[6]**Robert Ellis,*Lectures on topological dynamics*, W. A. Benjamin, Inc., New York, 1969. MR**0267561****[7]**Robert Ellis and Harvey Keynes,*A characterization of the equicontinuous structure relation*, Trans. Amer. Math. Soc.**161**(1971), 171–183. MR**0282357**, https://doi.org/10.1090/S0002-9947-1971-0282357-4**[8]**Walter Helbig Gottschalk and Gustav Arnold Hedlund,*Topological dynamics*, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R. I., 1955. MR**0074810****[9]**Edwin Hewitt and Kenneth A. Ross,*Abstract harmonic analysis. Vol. I*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR**551496****[10]**D. McMahon and T. S. Wu,*On proximal and distal extensions of minmal sets*, Bull. Inst. Math. Acad. Sinica**2**(1974), no. 1, 93–107. MR**0365539****[11]**Robert J. Sacker and George R. Sell,*Finite extensions of minimal transformation groups*, Trans. Amer. Math. Soc.**190**(1974), 325–334. MR**0350715**, https://doi.org/10.1090/S0002-9947-1974-0350715-8**[12]**William A. Veech,*Point-distal flows*, Amer. J. Math.**92**(1970), 205–242. MR**0267560**, https://doi.org/10.2307/2373504**[13]**W. A. Veech,*Almost automorphic functions on groups*, Amer. J. Math.**87**(1965), 719–751. MR**0187014**, https://doi.org/10.2307/2373071**[14]**-,*Generalizations of almost periodic functions*, Informal notes for lectures at Sir George Williams University, Montreal, Quebec, December 28-January 3, 1971.**[15]**Ta-sun Wu,*Homomorphisms in topological dynamics*, Hung-ching Chow Sixty-fifth Anniversary Volume, Math. Res. Center Nat. Taiwan Univ., Taipei, 1967, pp. 53–61. MR**0229227**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
54H20

Retrieve articles in all journals with MSC: 54H20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1976-0413067-2

Keywords:
Minimal,
transformation group homomorphism,
relativized equicontinuous structure relation,
connected

Article copyright:
© Copyright 1976
American Mathematical Society