This page gives information about one-day conference on Ergodic theory and set theory, the fourth in the series Set theory and its neighbours, which took place on Wednesday, 15th September 1999 at the London Mathematical Society building, De Morgan House, 57-58 Russell Square, London WC1.
The next meeting in the series, on Topology and set theory, will take place on Wednesday, 5th January, 2000.
The speakers at the meeting were
Here is the list of
participants.
What are the connections between ergodic theory and set theory?
Below is an extremely telegraphic list of some theorems
from the last few years showing some of the links. These theorems mostly
illustrate the influence of (descriptive) set theory on various
problems in ergodic theory.
Let G be the group of invertible measure preserving transformations on X a
standard measure space. Then (Halmos) G is a Polish
group. The ergodic transforamtions are a dense G_{\delta} set in G
(Halmos) while the strongly mixing maps are a meagre
F_{\sigma\delta} subset of G (Rohklin). The Bernoulli shifts are a \Pi^0_3
set in G. The entropy function is a Borel function, and
(Orenstein) two Bernoulli shifts are isomorphic iff they have the same
entropy, so that the conjugacy realtion on Bernoulli shifts is
smooth.
The isomorphism relation between ergodic discrete spectrum transformations
is inter-reducible to the identity relation on countable
sets of real numbers (Foreman/Louveau, using Halmos-von-Neumann in one
direction). E_0, (the equivalence relation of equal
modulo finite difference on sets of natural numbers, or equivalently of
equality mod the rationals on the reals) is Borel reducible to
the conjugacy relation on property K automorphisms of any fixed entropy
(Orenstein-Shields, Feldman), and so they are, eg, not
smoothly classifiable. The conjugacy relation on normal transformations of
a bounded order \alpha is Borel (Foreman), but the
conjugacy relation on all measure preserving transformations is complete
\Sigma^1_1 (Hjorth). There is a "turbulent" group
action on a Polish space such that the corresponding equivalence relation
is Borel reducible to norm 2 flows on T^2. So there is no
classification of norm 2 distal flows by countable structures (Hjorth).
The classs of measure distal transformations is a complete
\Pi^1_1 set, and the function that associates to each transformation the
least ordinal in an approximating tower is a \Pi^1_1 norm.
In particular every countable ordinal occurs as the least ordinal of an
approximating tower to some measure distal
transformation and the m.d.t.s of order bounded by \alpha are a Borel set
(Beleznay-Foreman).
More information about such issues will, we believe,
be found in a forthcoming expository paper by Matt Foremann, and in
a volume of papers, edited by Foreman and also forthcoming.
(We don't have any publication/preprint details but you might like to
try Foreman's website
over the next month or two to see if some pointers to such details appear.)
In the opposite direction) has been the use of notions from ergodic theory
in Hjorth et al's work on classifying equivalence
relations. See, for example, Hjorth's Classification and orbit equivalence
relations, and his talk at the first meeting in the Set
theory and its neighbours series. The paper Borel equivalence relations
and classifications of countable models, by Hjorth and
Kechris, Ann. Pure and Appl. Logic 82 (1996), 221-272, gives an
illustration of both tendencies.
Combinatorics
and set theory (the third meeting in the series), including
slides from the
talks and related preprints.
We hope to keep the meetings fairly relaxed, allowing plenty of
opportunity for informal discussion. We welcome and encourage anyone
to participate. Please do tell anyone about the meeting who you think
may be interested in it. And let us know if you would like to speak or have
ideas for speakers at future meetings. We would be grateful if you could
email us to let us know if you
intend to come, so that we can get a reasonable idea about number
of people likely to attend. Nevertheless you are very welcome simply
to turn up on the day if you make a late decision.
We have some limited funds to subsidise the travel
expenses of graduate students who would like to attend. Please
contact us for
details. We are very grateful to the LMS for allowing us to use De Morgan
House as a venue and for their financial support for these meetings. De
Morgan House is in the bottom left (i.e. south-east) corner of Russell
Square, itself in the bottom left hand corner of this map of the
area. The nearest tube
station is Russell Square, but De Morgan House is also only a short
walk from Euston, Euston Square and Goodge Street stations. We are very grateful to the British Logic Colloquium for financial support
for this meeting.
Last updated on 20th October 1999
Finite model
theory and set theory (the second meeting in the series), including
slides from the
talks and related preprints.
Set theory,
analysis and their neighbours (the first meeting in the series),
inlcuding
slides from the talks and related preprints.
