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A survey of minimal sets

Walter H. Gottschalk
1964 Annales de l'Institut Fourier  
The basic connection between almost periodic point and minimal set is this : If x e X then x is almost periodic if and only if the orbit closure xT of x is minimal [12] .  ...  In constructing nonequicontinuous minimal sets from coset transformation groups, the following theorem shows that it appears necessary to start from relatively complicated groups G: Let G be a locally  ... 
doi:10.5802/aif.160 fatcat:3ofaka4pmnhrdobzkvdooxt5wu

Relativized problems with abelian phase group in topological dynamics

D. McMahon
1976 Proceedings of the National Academy of Sciences of the United States of America  
Let (X, 7) be the equicontinuous minimal transformation group with X = irwZ7, the Cantor group, and S = z-Z2 endowed with the discrete topology acting on X by right multiplication.  ...  For any countable group Twe construct  ...  equicontinuous minimal set with abelian phase group be locally almost periodic?  ... 
doi:10.1073/pnas.73.4.1007 pmid:16592304 pmcid:PMC430187 fatcat:7jfm4bgp7zfjbf2buudti3jqp4

Minimal sets: An introduction to topological dynamics

W. H. Gottschalk
1958 Bulletin of the American Mathematical Society  
Hence X* is an almost periodic minimal set and thus a monothetic group, called the structure group of the original locally almost periodic minimal set.  ...  Since the monothetic groups can be constructed in their entirety as the character groups of the subgroups of the discrete circle group (Anzai and Kakutani 1943) [2] , the construction problem for almost  ... 
doi:10.1090/s0002-9904-1958-10223-2 fatcat:5nu3xkhtwrbm3px32ednqxs37m

Sturmian Minimal Sets

Gustav A. Hedlund
1944 American Journal of Mathematics  
We show the existence of such sets which contain asymptotic orbits, which are locally almost periodic and which are not only minimal under the defining transformation 7 but are also minimal under every  ...  The simplest example of a minimal set is a periodic motion. Minimal sets which are not periodic motions have been constructed by various devices (cf. e. g., Morse [1]).’  ... 
doi:10.2307/2371769 fatcat:tnfnrwkwxzef7pxmkmig4jtlke

Page 1309 of Mathematical Reviews Vol. 52, Issue 4 [page]

1976 Mathematical Reviews  
The trans- formation group (X,7) is almost periodic, regularly almost periodic, recursive if and only if (2*, 7) is, respectively, almost periodic, regularly almost periodic, recursive. L.  ...  Paul, Michael E. 9202 Construction of almost automorphic symbolic minimal flows. General Topology and Appl. 6 (1976), no. 1, 45-56.  ... 

Page 605 of American Journal of Mathematics Vol. 66, Issue 4 [page]

1944 American Journal of Mathematics  
We show the existence of such sets which contain asymptotic orbits, which are locally almost periodic and which are not only minimal under the defining transformation 7 but are also minimal under every  ...  The simplest example of a minimal set is a periodic motion. Minimal sets which are not periodic motions have been constructed by various devices (cf. e. g., Morse [1]).’  ... 

Page 1955 of Mathematical Reviews Vol. 58, Issue 3 [page]

1979 Mathematical Reviews  
Theorem 1: Let (7, X) be an effective almost periodic transformation group, where X is a compact, separable, locally Euclidean n-space; let J be a closed n-cell in X; then T(J) is locally connected.  ...  The author discusses various results of his and of H. Furstenberg on minimal sets in the context of transformation groups and, in particular, minimal transformation groups.  ... 

Page 127 of Mathematical Reviews Vol. 2, Issue 4 [page]

1941 Mathematical Reviews  
[MF 3016] A group G is called maximally (or minimally) almost periodic if there exists for every element a distinct from the unit element e (or for no element a) an almost periodic function f on G such  ...  Hence G; is an invariant subgroup of Gz generated by X and Y. M. S. Knebelman (Pullman, Wash.). v. Neumann, J. and Wigner, E. P. Minimally almost periodic groups.  ... 

Almost periodic functions, constructively

Bas Spitters, Anton Setzer
2005 Logical Methods in Computer Science  
Here we present a simple proof of Bohr's fundamental theorem for almost periodic functions which we then generalize to almost periodic functions on general topological groups.  ...  The almost periodic functions form a natural example of a non-separable normed space. As such, it has been a challenge for constructive mathematicians to find a natural treatment of them.  ...  Parts of this research can already be found in my PhD-thesis [9] . I would like to thank Wim Veldman for his advice during this period.  ... 
doi:10.2168/lmcs-1(3:3)2005 fatcat:5dhrmkjfxndvvkawe6qu4qtq4q

Extensions of minimal flows on manifolds

William Perrizo
1976 Proceedings of the American Mathematical Society  
Group extensions of vector field flows for which the orbit closure map forms a fiber bundle are constructed for the case of minimal flows on compact manifolds and compact Lie groups.  ...  Conditions for which minimal nontoral extensions exist are studied. on the left.  ...  Z and Y + aZ are pointwise almost periodic since both are G-lifts of pointwise almost periodic transformation groups.  ... 
doi:10.1090/s0002-9939-1976-0425933-8 fatcat:ws4x3atrj5at5hxush54nrsica

Extensions of Minimal Flows on Manifolds

William Perrizo
1976 Proceedings of the American Mathematical Society  
Group extensions of vector field flows for which the orbit closure map forms a fiber bundle are constructed for the case of minimal flows on compact manifolds and compact Lie groups.  ...  Conditions for which minimal nontoral extensions exist are studied. on the left.  ...  Z and Y + aZ are pointwise almost periodic since both are G-lifts of pointwise almost periodic transformation groups.  ... 
doi:10.2307/2041506 fatcat:wowwhfg7mjhb5i7omtc4nvkhuu

Page 664 of Mathematical Reviews Vol. 35, Issue 3 [page]

1968 Mathematical Reviews  
Lelek (Warsaw) Wu, Ta-sun 3650 Construction of locally almost periodic minimal trans- formation groups. Math. Systems Theory 1 (1967), 157-163. The author considers the following problem.  ...  Is there a topology » on G x J and an action of T on GxJ such that (1) (@xJ, ») isa compact Hausdorff space, (2) (@ x J, 7') is a locally almost periodic minimal transformation group, (3) the map (g,a)  ... 

The fundamental group and the first cohomology group of a minimal set

Hsin Chu, Michael A. Geraghty
1963 Bulletin of the American Mathematical Society  
It is further shown that for such spaces X a totally minimal flow cannot be locally almost periodic. So on such spaces one cannot have a locally almost periodic minimal flow.  ...  almost periodic.  ...  Denoting by P the proximal relation, since X is locally almost periodic we have that P is a closed equivalence relation (see [3] ) and the induced transformation group (X/P, T, 7T*) is almost periodic  ... 
doi:10.1090/s0002-9904-1963-10933-7 fatcat:3pioaaj4fzcepb55lgckqcuqia

Embedding a transformation group in an automorphism group

Murray Eisenberg
1969 Proceedings of the American Mathematical Society  
Leonard Shapiro has kindly pointed out that a proposition equivalent to our lemma appears in a paper of R. Ellis  ...  Using a construction of Baidosov [l], we show that a topological transformation group with completely regular phase space X and locally compact phase group T can be equivariantly embedded in a transformation  ...  The almost periodicity of (x, y) now follows from that of x*y by means of the lemma.  ... 
doi:10.1090/s0002-9939-1969-0244979-2 fatcat:kz6izdkm4za5napsy5v4ijdily

Page 281 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 94, Issue 2 [page]

1960 American Mathematical Society. Transactions of the American Mathematical Society  
Theorem 3 and its corollaries indicate that given a compact group I, a dense subgroup 7, and a set J one should be able to construct the minimal ideal J of a locally almost periodic transformation group  ...  An interesting problem is to provide 'XJ with a compact Hausdorff topology and to determine the action of T on !XJ such that (T XJ, T) is locally almost periodic. REFERENCES 1.  ... 
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