Re: Few questions about topological groups

jordilluis@xxxxxxxxx wrote:
Hello,

i've a question about topological groups. Let's start by a
definition:

Let (G,*) be a topological group and let g\in G. We will consider G
to be compact. We will denote by e the identity element.

We define the orbit of g relative to h\in G as \{g^n*h\}_{\forall n
\in \mathbb{Z}}, (with \mathbb{Z} the integers numbers.) We will
denote this by O(g,h).

Question1: For each topological group G, is there a pair (g,h) which
O(g,h) is dense in G? (i.e. the orbit of h under the iterates of g is
dense).

Question2: For each topological group G, is there g such that O(g,e)
is dense in G? (i.e. the orbit of the identity element under the
iterates of g is dense).

This two results are well knowed when G is the n-torus.

Thank you very much,

I found the following reference (among others) when I Googled
the phrase

topologically cyclic

http://books.google.com/books?id=AfBzWL5bIIQC&pg=PA177&;
lpg=PA177&dq=topologically+cyclic&source=web&ots=U9emCdf7yf
&sig=QFnl01kYlme8cgvh-aLnpP6R6vc


You'll have to paste the three lines together to get the full
URL, or just do the same Google search I did.

Here's a segment of the as provided by Google:

A group is called topologically cyclic if it contains
an element, called a generator, whose powers are dense
in the group. ...

The reference is to the text

Representations of Compact Lie Groups
By Theodor Bröcker, Tammo tom Dieck

and the above text is followed by this remark:

Recall that a compact Lie group is topologically
cyclic precisely if it is isomorphic to the product
of a torus and a finite cyclic group.

Unfortunately, that is actually more than I know on the topic,
so I'll just have to leave you to your own devices.

Dale
.

Relevant Pages

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    ... For each topological group G, is there g such that Ois dense in G? ... (i.e. the orbit of the identity element under the iterates of g is dense). ... The topology is the discrete topology. ...
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  • Re: Few questions about topological groups
    ... For each topological group G, is there a pair which Ois dense in G? ... (i.e. the orbit of the identity element under the iterates of g is dense). ...
    (sci.math)
  • Few questions about topological groups
    ... For each topological group G, is there g such that Ois dense in G? ... (i.e. the orbit of the identity element under the iterates of g is dense). ... This two results are well knowed when G is the n-torus. ...
    (sci.math)
  • Re: Few questions about topological groups
    ... We will denote by e the identity element. ... Ois dense in G? ... (i.e. the orbit of h under the iterates of g is ...
    (sci.math)