Talk:Krylov-Bogolyubov theorem
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[edit] Found on newsgroup
The following excerpt from sci.math may be of interest to future editors.
The Relationship Between Measure-Theoretic and Topological Notions:
It does not make sense to say that ergodicity is EQUIVALENT to TT (topological
transitivity). A topological dynamical system, i.e. X compact metric with
T: X \rightarrow X continuous, has a different structure from a measure
theoretical dynamical system, i.e., (X, \M, \mu) a probability measure space
with T:X \rightarrow X a measureable mapping preserving the \mu measure, in the
sense that for all measureable sets E, \mu T^{-1}(E) = \mu (E).
However, given a TDS (topological dynamical system), there are always MTDS's
(measure-theoretical dynamical systems) associated with it. First, take \M
to be the Borel algebra (smallest sigma algebra containing all the open and
closed sets). Then the Krylov-Bogoliubov Theorem says that every T
continuous has at least one T-invariant measure, i.e. \mu such that
(X, Borel, \mu, T) is a measure theoretical dynamical system. Moreover,
the set of Borel measures on X form a compact convex subset of a separable
Banach space, an infinite dimensional ``simplex'' whose vertices correspond
to delta measures on X, and for each T, the set of T-invariant measures
forms a ``sub-simplex''. (You can actually draw this for the case of X
a finite space with the discrete topology; the geometry is quite beautiful,
although rather misleading vis a vis infinite X.) A beautiful theorem says
that the ergodic T-invariant measures are exactly the vertices of the
sub-simplex of T-invariant measures! The point is this:
:EVERY TDS is associated with at least one ergodic MTDS!
Also, TDS's clearly have ``a higher level of structure'' than MTDS's,
so your assertion about TT and ergodicity confused two levels of
structure.
[Note: A delta measure \delta_x is indeed a point mass: for all Borel
measureable sets E, \delta_x(E) = 1 if x \in E and 0 otherwise.]
Nevertheless, there are relations between topological and measure theoretic
conditions. For instance, if (X,T) is an invertible TDS (T is a homeo),
and if there exists an ergodic T-invariant measure which gives positive
measure to every nonempty open set, then (X, T) is TT.
Incidentally, the spelling “Bogoliubov” gets somewhat more Google hits. --KSmrqT 00:17, 27 October 2006 (UTC)
Noted. Krylov-Bogoliubov theorem redirects to the main article, Krylov-Bogolyubov theorem. Sullivan.t.j 10:07, 27 October 2006 (UTC)
[edit] Possible source to cite?
The Springer Online Encyclopædia of Mathematics has an article on the Krylov–Bogolyubov method of averaging which may be of interest. --KSmrqT 03:24, 27 October 2006 (UTC)
